Quantum Performance of Thermal Machines over Many Cycles

The performance of quantum heat engines is generally based on the analysis of a single cycle. We challenge this approach by showing that the total work performed by a quantum engine need not be proportional to the number of cycles. Furthermore, optimizing the engine over multiple cycles leads to the identification of scenarios with a quantum enhancement. We demonstrate our findings with a quantum Otto engine based on a two-level system as the working substance that supplies power to an external oscillator.

Figure 1

Schematic quantum heat engine. The quantum engine $E$ does work $w$ on an external system $S$ through the coupling $H_{SE}$ absorbing heat $Q$ from the baths collectively represented by $B$, which consists of hot ($B_1$) and cold ($B_2$) baths.

Figure 2

Quantum performance of a heat engine. (a) Schematic setup of a TLS engine ($E_{\mathrm{ad}}$ being the adiabatic energy levels) running with a hot ($B_1$) and cold ($B_2$) bath and coupled to a HO system $H_S$ via $H_{SE}$. Average work ${⟨w⟩}_n$ and ${⟨\tilde{w}⟩}_n$ as functions of the number of cycles $n$ for the perturbative ($g=0.02$) and impulse-type coupling [(b)] and for the nonperturbative ($g=0.5$, $\alpha T=2142$) and continuous coupling [(c)]. Lines in (b) and (c) are from numerical calculations, and dots in (b) are from the analytical expressions (9) and (10). (d) $g_{SE}(t)$ (cyan dashed line) and ${⟨{\sigma }_x^{(I)}(t)⟩}_{{\rho }_0^{EB}}$ (red solid line) for the first cycle in the case of (c). The marked difference in the dynamics of ${\sigma }_x(t)$ in the two strokes of the engine comes from the interaction of the engine with the cold bath at $t=0$ and the hot one at $t=T/2$ leading to an almost pure state at $t=0$ and an almost mixed state at $t=T/2$ for the present choice of parameters. We set $b=0.1/\mathrm{\Delta }$ in (b) and ${\delta }_t=0.98$ in (c) and (d). Other parameters are $\omega T=0.05\times 2\pi$, $v=0.5{\mathrm{\Delta }}^2$, $T=20/\mathrm{\Delta }$, ${\beta }_h=1/4E_{\max }$, and ${\beta }_c=1/\mathrm{\Delta }$.

Figure 3

Quantum work statistics. Comparison of the probability distribution functions $p$’s of work (a) $w$ and (b) $\tilde{w}$ obtained in our numerical calculations for the quantum engine shown in Fig. 2 and those for a classical force $f$ in Eq. (12). Here, $p$’s after 20 cycles are shown. The inset in (a) compares ${⟨w⟩}_n$ performed by a quantum engine and by a classical force.