Finite-time quantum Otto engine: Surpassing the quasistatic efficiency due to friction

In finite-time quantum heat engines, some work is consumed to drive a working fluid accompanying coherence, which is called “friction.” To understand the role of friction in quantum thermodynamics, we present a couple of finite-time quantum Otto cycles with two different baths: Agarwal versus Lindbladian. We solve them exactly and compare the performance of the Agarwal engine with that of the Lindbladian engine. In particular, we find remarkable and counterintuitive results that the performance of the Agarwal engine due to friction can be much higher than that in the quasistatic limit with the Otto efficiency, and the power of the Lindbladian engine can be nonzero in the short-time limit. Based on additional numerical calculations of these outcomes, we discuss possible origins of such differences between two engines and reveal them. Our results imply that, even with an equilibrium bath, a nonequilibrium working fluid brings on the higher performance than what an equilibrium working fluid does.

Figure 1

A finite-time quantum Otto cycle is schematically illustrated with harmonic potentials and Wigner functions, which consists of isochoric and adiabatic processes. In the isochore, the working fluid exchanges heat with heat bath of temperature $T_{\mathrm{h}}\left(T_{\mathrm{c}}\right)$ by the propagator ${\mathcal{P}}_{\mathrm{h}}\left({\mathcal{P}}_{\mathrm{c}}\right)$ of the vector $(\langle \widehat{H}\rangle ,\langle \widehat{L}\rangle ,\langle \widehat{D}\rangle ,\langle \widehat{I}\rangle )$ for the process time ${\tau }_{\mathrm{h}}\left({\tau }_{\mathrm{c}}\right)$, whereas, in the adiabatic expansion (compression), the internal energy change of the working fluid becomes work by ${\mathcal{P}}_{\mathrm{hc}}\left({\mathcal{P}}_{\mathrm{ch}}\right)$ for ${\tau }_{\mathrm{hc}}\left({\tau }_{\mathrm{ch}}\right)$. The total-energy expectation of the working fluid is drawn as a function of $\omega$ in the middle panel.

Figure 2

The contour plots of ${\eta }^{{}_{\mathbf{A}}}$ (upper left), ${\eta }^{{}_{\mathbf{L}}}$ (upper right), ${\eta }^{{}_{\mathbf{A}}}/{\eta }_{\mathrm{O}}$ (lower left) and ${\eta }^{{}_{\mathbf{L}}}/{\eta }_{\mathrm{O}}$ (lower right) are presented as a function of ${\tau }_{\mathrm{h}}+{\tau }_{\mathrm{c}}$ (the sum of the isochoric time, $y$ axis) and ${\tau }_{\mathrm{ch}}+{\tau }_{\mathrm{hc}}$ (the sum of the adiabatic time, $x$ axis). Since we plot only when the cycle behaves as a heat engine, there are diagonal blank spaces. Unless isochoric process time is long, the blank spaces well coincide with purple dotted (green dashed) lines which are derived from Eq. (24) when $n$ is odd (even). Near purple resonance lines, we can find some regions that show ${\eta }^{{}_{\mathbf{A}}}\gg {\eta }^{{}_{\mathbf{L}}}$. In both panels, the red solid line represents $({\tau }_{\mathrm{ch}}+{\tau }_{\mathrm{hc}})/({\tau }_{\mathrm{h}}+{\tau }_{\mathrm{c}})=1/5$, and the orange dot corresponds to the case of ${\tau }_{\mathrm{cyc}}=1.2$, which is discussed in Figs. 3 and 4. Here we set all parameters to be dimensionless and $\hslash =k_B=m=\gamma =1$, ${\omega }_{\mathrm{h}}=4$, ${\omega }_{\mathrm{c}}=3$, $T_{\mathrm{h}}=200$, and $T_{\mathrm{c}}=1$, which are kept used from now on unless other values are indicated separately. For the simplicity, we choose ${\tau }_{\mathrm{h}}={\tau }_{\mathrm{c}}$ and ${\tau }_{\mathrm{ch}}={\tau }_{\mathrm{hc}}$.

Figure 3

Along the red line of each panel in Fig. 2, we compare the performance of the Agarwal Otto engine (blue, $•$) with that of the Lindbladian (red, $•$), in the context of its efficiency $\eta$ (top), power $P$ (middle), and entropy $S$ (bottom), which are functions of ${\tau }_{\mathrm{cyc}}$. In the quasistatic limit (${\tau }_{\mathrm{cyc}}\to \infty$), ${\eta }^{{}_{\mathbf{A}}},{\eta }^{{}_{\mathbf{L}}}\to {\eta }_{{}_{\mathrm{O}}}$, and $P^{{}_{\mathbf{A}}},P^{{}_{\mathbf{L}}}\to 0$, which are drawn as horizontal red solid lines. In addition, the analytic short-time results of Eq. (25) for ${\eta }^{{}_{\mathbf{L}}}$ and $P^{{}_{\mathbf{L}}}$ are drawn as horizontal black dashed lines up to ${\tau }_{\mathrm{cyc}}=1.2$, whereas ${\eta }^{{}_{\mathbf{A}}}\to 0$ and $P^{{}_{\mathbf{A}}}\to 0$. Vertical orange solid lines are drawn at ${\tau }_{\mathrm{cyc}}=1.2$ (orange dots in Fig. 2), where ${\eta }^{{}_{\mathbf{A}}}>{\eta }_{\mathrm{O}}>{\eta }^{{}_{\mathbf{L}}}$. For nonengine or unphyiscal values, we use different symbols from that of the heat engine and put some explanations as keys: fridge, useless, and divergent.

Figure 4

At ${\tau }_{\mathrm{cyc}}=1.2$ when ${\eta }_{{}_{\mathrm{A}}}>{\eta }_{{}_{\mathrm{O}}}$ (indicated in Fig. 3), the expectation values of the kinetic energy (KE, dashed lines) and the potential energy (PE, solid lines) are plotted as a function of time $t$, where we set ${\tau }_{\mathrm{h}}=0.1$ and ${\tau }_{\mathrm{hc}}=0.5$. Three vertical black solid lines represent three boundaries, from the adiabatic compression to the hot isochore, from the hot isochore to the adiabatic expansion, and from the adiabatic expansion to the cold isochore, respectively (from left to right). Here we use the same parameters and colors as used in Fig. 3. In Agarwal's adiabatic expansion (the last part for $0.7\le t\le 1.2$, see the two blue lines), the PE is always larger than the KE, which is different from the Lindbladian where the sign of the Lagrangian changes. This implies that the friction term $-\frac{\dot{\omega }\left(t\right)}{\omega \left(t\right)}\langle \widehat{L}\rangle$ in Eq. (13) contributes to ${\eta }^{{}_{\mathbf{A}}}>{\eta }_{{}_{\mathrm{O}}}$.

Figure 5

The contour plots of ${\eta }^{{}_{\mathbf{A}}}$ (upper left), ${\eta }^{{}_{\mathbf{L}}}$ (upper right), ${\eta }^{{}_{\mathbf{A}}}/{\eta }_{\mathrm{O}}$ (lower left), and ${\eta }^{{}_{\mathbf{L}}}/{\eta }_{\mathrm{O}}$ (lower right) are shown as a function of $T_{\mathrm{c}}/T_{\mathrm{h}}$ (temperature ratio, $x$ axis) and ${\omega }_{\mathrm{c}}/{\omega }_{\mathrm{h}}$ (frequency ratio, $y$ axis). Here most parameters are the same as before, but we change ${\omega }_{\mathrm{c}}=3, T_{\mathrm{c}}=100, {\tau }_{\mathrm{c}}={\tau }_{\mathrm{h}}=2$, and ${\tau }_{\mathrm{ch}}={\tau }_{\mathrm{hc}}=0.4$, which nicely show how our enumeration results are bounded by the condition derived in the high-temperature limit. Blue (orange) guided lines are the boundaries between the heat engine and the others such as the refrigerator and the useless machine in the short-time (quasistatic) limit. The short-time limit was obtained from the Lindbladian work expression of Eq. (25) in the high-temperature limit. When the frequency of the harmonic oscillator gets higher, the approximation of the short-time limit cannot be valid anymore. Therefore, ${\eta }^{{}_{\mathrm{A}}}$ and ${\eta }^{{}_{\mathrm{L}}}$ near the small frequency ratio can be in between the blue lines and the orange lines. The insets correspond to the quasistatic limit, where both cases show the same results.