Non-Markovian effect on quantum Otto engine: Role of system-reservoir interaction

We study a limit cycle of a quantum Otto engine whose every cycle consists of two finite-time quantum isochoric (heating or cooling) processes and two quantum adiabatic work-extracting processes. Considering a two-level system as a working substance that weakly interacts with two reservoirs comprising an infinite number of bosons, we investigate the non-Markovian effect [short-time behavior of the reduced dynamics in the quantum isochoric processes (QIPs)] on work extraction after infinite repetition of the cycles. We focus on the parameter region where energy transferred to the reservoir can come back to the system in a short-time regime, which we call energy backflow to show partial quantum-mechanical reversibility. As a situation completely different from macroscopic thermodynamics, we find that the interaction energy is finite and negative by evaluating the average energy change of the reservoir during the QIPs by means of the full-counting statistics, corresponding to the two-point measurements. This feature leads us to the following findings: (1) The Carnot theorem is consistent with a definition of work including the interaction energy, although the commonly used definition of work excluding the interaction leads to a serious conflict with the thermodynamic law, and (2) the energy backflow can increase the work extraction. Our findings show that we need to pay attention to the interaction energy in designing a quantum Otto engine operated in a finite time, which requires us to include the non-Markovian effect, even when the system-reservoir interaction is weak.

Figure 1

Our protocol of the QOE. First QIP ($A_{+}\to B_{-}$): The system with the level difference ${\omega }_{\mathrm{h}}$ contacts the reservoir at $T_{\mathrm{h}}$ during $t_1$, exchanging energy. This is achieved by switching ${\gamma }_{\mathrm{h}}\left(t\right)$ to be set to unity and after time $t_1$ set back to zero, while the difference in level ${\omega }_0\left(t\right)={\omega }_{\mathrm{h}}$ is maintained. First QAP ($B_{+}\to C_{-}$): We decrease ${\omega }_0\left(t\right)$ quantum adiabatically from ${\omega }_{\mathrm{h}}$ to ${\omega }_{\mathrm{c}}$ maintaining the population of each level to transfer energy from the system to external working storage, which may be realized by adiabatically expanding the box containing the two-level system. Second QIP ($C_{+}\to D_{-}$): The system with ${\omega }_{\mathrm{c}}$ contacts the reservoir at $T_{\mathrm{c}}$ during $t_2$. Second QAP ($D_{+}\to A_{-}$): We increase ${\omega }_0\left(t\right)$ quantum adiabatically from ${\omega }_{\mathrm{c}}$ back to ${\omega }_{\mathrm{h}}$, which may be realized by compressing the box by using energy stored in the external working storage. The dots on the levels in the two-level system schematically represent the ratio of populations of the upper and lower states.

Figure 2

Non-Markovian (NM; solid line) and Markovian (M; dashed line) dynamics of (a) the energy flow, (b) the effective temperature, and (c) the change in energy of respective parts of the engine during contact with a hot reservoir; a flow to the reservoir is defined to be positive. We find that (a) the non-Markovian dynamics shows the energy backflow from the reservoir and (b) the effective temperature of the system becomes higher than the hot reservoir. In (c), we show the change in energy, $\mathrm{\Delta }{E}_{\mathrm{S}}^{\mathrm{h}}$, $\mathrm{\Delta }{E}_{\mathrm{B}}^{\mathrm{h}}$, and $E_{\mathrm{I}}^{\mathrm{h}}$. We find that $E_{\mathrm{I}}^{\mathrm{h}}$ is finite in the non-Markovian dynamics. The parameter settings are $T_{\mathrm{h}}=5.0$, $T_{\mathrm{c}}=1.0$, $t_2=60$, $\lambda =0.01$, $\mathrm{\Omega }=0.4$, and ${\omega }_{\mathrm{c}}/{\omega }_{\mathrm{h}}=0.18$ for unit values ${\omega }_{\mathrm{h}}=1$ with $k_{\mathrm{B}}=\hslash =1$. If an actual value of the level splitting of the two-level system in contact with the hotter reservoir is $\hslash {\omega }_{\mathrm{h}}=1$ $\mathrm{meV}$, the actual values of the other parameters are roughly evaluated as follows: $\lambda =10$ $\mathrm{meV}$, $\hslash \mathrm{\Omega }=0.4$ $\mathrm{meV}$, $T_{\mathrm{h}}\approx 58$ $\mathrm{K}$, $T_{\mathrm{c}}\approx 11.8$ $\mathrm{K}$, and $\hslash {\omega }_{\mathrm{c}}=0.18$ $\mathrm{meV}$.

Figure 3

Extracted work $W_{\mathrm{II}}$, Eq. (12), of the QOE under the non-Markovian dynamics for contact durations $t_1$ and $t_2$ with the detachment energy taken into account. The parameter settings are $T_{\mathrm{h}}=5.0$, $T_{\mathrm{c}}=1.0$, $\lambda =0.01$, $\mathrm{\Omega }=0.4$, and ${\omega }_{\mathrm{c}}/{\omega }_{\mathrm{h}}=0.18$ for unit values ${\omega }_{\mathrm{h}}=1$ with $k_{\mathrm{B}}=\hslash =1$ (same as in Fig. 2).

Figure 4

The work extraction from the QOE with isochoric processes evaluated under the Born-Markov approximation for the contact durations $t_1$ and $t_2$ with hotter and colder reservoirs, respectively. We set the parameters to $T_{\mathrm{h}}=5.0$, $\lambda =0.01$, $\mathrm{\Omega }=0.4$, and ${\omega }_{\mathrm{c}}/{\omega }_{\mathrm{h}}=0.18$ under the unit values of ${\omega }_{\mathrm{h}}=1$ with $k_{\mathrm{B}}=\hslash =1$, for which ${\eta }_{\mathrm{O}}=0.82>{\eta }_{\mathrm{C}}=0.8$.

Figure 5

Work extracted $W_{\mathrm{I}}$ of the QOE under the non-Markovian dynamics for contact durations $t_1$ and $t_2$ evaluated by using the first definition of work [Eq. (8)]. For settings corresponding to ${\eta }_{\mathrm{O}}>{\eta }_{\mathrm{C}}$, we find a regime $t_1\lesssim 5$ where $\langle W\rangle$ is positive. The parameter settings are $T_{\mathrm{h}}=5.0$, $T_{\mathrm{c}}=1.0$, $\lambda =0.01$, $\mathrm{\Omega }=0.4$, and ${\omega }_{\mathrm{c}}/{\omega }_{\mathrm{h}}=0.18$ for unit values ${\omega }_{\mathrm{h}}=1$ with $k_{\mathrm{B}}=\hslash =1$ (same as in Fig. 2).

Figure 6

Dependence of the work extraction on ${\omega }_{\mathrm{h}}$ and ${\omega }_{\mathrm{c}}$ evaluated under non-Markovian dynamics for the contact durations $t_1$ and $t_2$ with hotter and colder reservoirs, respectively. We set the parameters to $T_{\mathrm{h}}=5.0$, $\lambda =0.01$, and $\mathrm{\Omega }=0.4$ under the unit values of $k_{\mathrm{B}}=\hslash =1$.

Figure 7

Dependence of the work extraction on ${\omega }_{\mathrm{h}}$ and ${\omega }_{\mathrm{c}}$ evaluated under non-Markovian dynamics for the contact duration $t_1$ and $t_2$ with the detachment energy counted in. We set the parameters to $T_{\mathrm{h}}=5.0$, $\lambda =0.01$, and $\mathrm{\Omega }=0.4$ under the unit values of $k_{\mathrm{B}}=\hslash =1$.

Figure 8

Non-Markovian (NM; solid line) and Markovian (M; dashed line) dynamics of (a) the energy flow, (b) the effective temperature, and (c) the change in energy of respective parts of the engine during contact with a hot reservoir for a large value of the cutoff frequency $\mathrm{\Omega }=3.0$. The other parameters are set to $T_{\mathrm{h}}=5.0$, $T_{\mathrm{c}}=1.0$, $t_2=60$, $\lambda =0.01$, and ${\omega }_{\mathrm{c}}/{\omega }_{\mathrm{h}}=0.18$ for unit values ${\omega }_{\mathrm{h}}=1$ with $k_{\mathrm{B}}=\hslash =1$.

Figure 9

Work extracted $W_{\mathrm{I}}$ of the QOE under the non-Markovian dynamics for contact durations $t_1$ and $t_2$ evaluated by using the conventional definition of work given by Eq. (8) for a large cutoff frequency $\mathrm{\Omega }=3.0$. The other parameters are set to $T_{\mathrm{h}}=5.0$, $T_{\mathrm{c}}=1.0$, $\lambda =0.01$, and ${\omega }_{\mathrm{c}}/{\omega }_{\mathrm{h}}=0.18$ for unit values ${\omega }_{\mathrm{h}}=1$ with $k_{\mathrm{B}}=\hslash =1$.