Speed and efficiency limits of multilevel incoherent heat engines

We present a comprehensive theory of heat engines (HE) based on a quantum-mechanical “working fluid” (WF) with periodically modulated energy levels. The theory is valid for any periodicity of driving Hamiltonians that commute with themselves at all times and do not induce coherence in the WF. Continuous and stroke cycles arise in opposite limits of this theory, which encompasses hitherto unfamiliar cycle forms, dubbed here hybrid cycles. The theory allows us to discover the speed, power, and efficiency limits attainable by incoherently operating multilevel HE depending on the cycle form and the dynamical regimes.

Figure 1

(a) Schematic view of the generic HE: multilevel system $S$ with periodic frequency modulation $\omega (k,t)$ and amplitude-modulation of the couplings $f_{\mathrm{h}}(k,t),f_{\mathrm{c}}(k,t)$ to thermal baths $B_{\mathrm{h}}$ (red), $B_{\mathrm{c}}$ (blue) with distinct spectra. (b) Frequency modulation $\omega (k,t)$: time-variation for different $k$ from continuous to Otto cycles (see text). (c) $f_{\mathrm{h}}(k,t)$ (red) and $f_{\mathrm{c}}(k,t)$ (blue) as a function of time for $k=0$ (solid line), $k=2$ (dotted line), and $k=100$ (dashed line) (see text).

Figure 2

(a) Spectral representation of the hot and cold baths and frequency modulation sidebands in the Markovian limit. (b) Heat currents and power for the continuous cycle $\left(k=0\right)$ yield two possible regimes: HE ($\overline{\dot{W}}<0,{\overline{J}}_{\mathrm{c}}<0,{\overline{J}}_{\mathrm{h}}>0$) for ${\mathrm{\Delta }}_{\mathrm{m}}<{\mathrm{\Delta }}_{\mathrm{cr}}$ and refrigerator (REF) ($\overline{\dot{W}}>0,{\overline{J}}_{\mathrm{c}}>0,{\overline{J}}_{\mathrm{h}}<0$) for ${\mathrm{\Delta }}_{\mathrm{m}}>{\mathrm{\Delta }}_{\mathrm{cr}}$ [cf. Eq. (16)]. For our choice of parameters, ${\mathrm{\Delta }}_{\mathrm{cr}}={\omega }_0/2$. (c) Same for a hybrid cycle $\left(k=2\right)$. Now we have three regimes: HE, heat distributor (HD) ($\overline{\dot{W}}>0,{\overline{J}}_{\mathrm{c}}<0,{\overline{J}}_{\mathrm{h}}\in \mathbb{R}$), and refrigerator. Here $T_{\mathrm{c}}=10,T_{\mathrm{h}}=30,{\omega }_0=3,\lambda =0.1,G_{\mathrm{c}}(0<{\omega }_0-q{\mathrm{\Delta }}_{\mathrm{m}}<{\omega }_0)=G_{\mathrm{h}}({\omega }_0+q{\mathrm{\Delta }}_{\mathrm{m}}>{\omega }_0)=1$. (d) Schematic operational regimes of HE, heat distributor, and refrigerator.

Figure 3

A continuous cycle (red dot) is obtained for $k=0$, whereas $k\to \infty$ describes a realistic Otto cycle. For a hybrid cycle, the heat machine operates in the heat engine, heat distributor or refrigerator regimes depending on the modulation rate ${\mathrm{\Delta }}_{\mathrm{m}}$. The heat distributor regime is absent for the continuous cycle $k=0$.

Figure 4

(a) Efficiency $\eta$ scaled by ${\eta }_{\mathrm{Carnot}}$ as a function of modulation rate ${\mathrm{\Delta }}_{\mathrm{m}}$ for the different $k$ values. $\eta (k=0)$ may exceed the Curzon-Ahlborn bound, as well as the Otto-cycle efficiency ${\eta }_{\mathrm{Otto}}$, whereas cycles with larger $k$ fail to do so. Our choice of parameters yields ${\mathrm{\Delta }}_{\mathrm{cr}}\approx 0.98{\omega }_0$, thus signifying that for the continuous $k=0$ cycle, the WF operates in the refrigerator regime only for very large ${\mathrm{\Delta }}_{\mathrm{m}}$, not visible in the range plotted above. (b) Speed limit: ${\left({\mathrm{\Delta }}_{\mathrm{m}}\right)}_{\mathrm{SL}}$ as a function of $k$, same parameters as in (a). Continuous-cycle limit: red dot. (c): Efficiency $\eta$ scaled by ${\eta }_{\mathrm{Carnot}}$ as a function of $k$ (same parameters as above): $\eta$ tends to ${\eta }_{\mathrm{Otto}}$ as ${\mathrm{\Delta }}_{\mathrm{m}}$ decreases. Red dots show the efficiencies for the continuous-cycle. (Here ${\omega }_1=1,{\omega }_2=5,{\omega }_0=3,T_{\mathrm{c}}=1,T_{\mathrm{h}}=100,G_{\mathrm{c}}(0<{\omega }_0-q{\mathrm{\Delta }}_{\mathrm{m}}<{\omega }_0)=G_{\mathrm{h}}({\omega }_0+q{\mathrm{\Delta }}_{\mathrm{m}}>{\omega }_0)=1$).

Figure 5

Absolute generated average power $|\overline{\dot{W}}|$ scaled by $\delta {\omega }_{\max }$ as a function of $k$. $\delta {\omega }_{\max }\left(k\right)$ is the maximum modulation amplitude of $\omega (k,t)$, which is related to the input power. A hybrid HE cycle $\left(k\approx 1.1\right)$ yields a higher power than either a continuous or a realistic Otto cycle with $k\to \infty$. The continuous-cycle limit is shown by the red dot. (Here ${\mathrm{\Delta }}_{\mathrm{m}}=1.2$) Inset: Same as a function of ${\mathrm{\Delta }}_{\mathrm{m}}$. In the limit of ${\mathrm{\Delta }}_{\mathrm{m}}\to 0$ (and ${\tau }_{\mathrm{I}}\to \infty$), the average power vanishes for large $k$, as indicated in the plot. (Same parameters as in Fig. 4.)

Figure 6

${\beta }_{\mathrm{eff}}/{\beta }_{\mathrm{c}}$ plotted as a function of time for $k=1$ (solid line) and $k=20$ (thin dotted line). ${\beta }_{\mathrm{eff}}$ changes abruptly during the unitary stroke for $k=20$, even though $f_h\left(t\right)$ (thick dashed line) $\approx 0$ for $t<{\tau }_{\mathrm{I}}/4$ (see red line for the discrepancy), showing that ${\beta }_{\mathrm{eff}}$ loses its meaning in the limit of large $k$. By contrast, ${\beta }_{\mathrm{eff}}$ varies smoothly for $k=1$ (solid line). (Same parameters as in Figs. 4, 5, and ${\mathrm{\Delta }}_{\mathrm{m}}=1.4$).