Quantum Thermal Machines Improved by Internal Coupling: From Equilibrium to Non-equilibrium Limit Cycles

Abstract

We investigate how internal coupling influences the operation and performance of a quantum Otto cycle operating as the Gibbs-state limit cycle (GSLC), equilibrating limit cycle (ELC), and non-equilibrating limit cycle (NELC). We show that the internal coupling significantly broadens the operational regime of the cycle. In particular, in parameter regimes where the uncoupled Otto cycle fails to operate as any thermal machine, the coupled system can function as an engine or a refrigerator. For the GSLC, in which we assume that the system quickly equilibrates during the isochoric processes, the internal coupling not only shifts and enlarges the operational regime but also enhances the efficiency and the coefficient of performance (COP), allowing the performance to exceed the standard Otto bounds while remaining below the Carnot limit. For ELC and NELC, we validate the global approach of the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) master equation by comparison with the GSLC, and examine the NELC for finite interaction time and the ELC for infinite interaction time. Although the efficiency and COP of NELC are lower than those of ELC, shorter interaction times yield higher power output, consistent with the power--efficiency trade-off.

Figures (12)

• Figure 1: Diagram of the system evolution of the (a) Gibbs-state limit cycle (GSLC), (b) equilibrating limit cycle (ELC), and (c) non-equilibrating limit cycle (NELC). For the GSLC in the panel (a), we do not consider the influence of the interaction time and assume that the system equilibrates rapidly in each isochoric process. Considering the Otto cycle as an open quantum system, it takes time for the system to equilibrate in each isochoric process. When the interaction time is short, the system state cannot reach the steady state in each isochoric process, but converges to the NELC as in panel (c). After the interaction time approaches infinity, the system equilibrates in each isochoric process and operates as the ELC in the panel (b).
• Figure 2: Schematic view of quantum Otto cycle without any internal couplings. The internal system is a two-level system with excited energy level alternates between $\omega_{\mathrm{h}}$ and $\omega_{\mathrm{c}}$. In the isochoric processes (a) and (c), it interacts with heat baths with the inverse temperature $\beta_{\mathrm{h}}$ and $\beta_{\mathrm{c}}$, respectively. In the adiabatic processes (b) and (d), it interacts with the work storage denoted by $W$. We assume that the interaction time in the adiabatic processes is negligible.
• Figure 3: Schematic view of quantum Otto cycle with the internal coupling between the excited and ground energy levels of the internal two-level system. The internal coupling strength is $g_{\mathrm{h}}$ and $g_{\mathrm{c}}$ in the isochoric process (a) and (c), respectively. In the adiabatic processes (b) and (d), not only do the excited energy levels alternate between $\omega_{\mathrm{h}}$ and $\omega_{\mathrm{c}}$, but also the internal coupling strengths alternate between $g_{\mathrm{h}}$ and $g_{\mathrm{c}}$.
• Figure 4: Logarithmic plots of the energy flows (a) $\log|Q_{\mathrm{h}}|$, (b) $\log|Q_{\mathrm{c}}|$, and (c) $\log|W|$ depending on $g_\alpha/\omega_\alpha$ evaluated in the Gibbs-state limit cycle (GSLC). The black boundaries indicate the zero energy flow. The symbol P indicates the areas of the positive energy flow, in which the system absorbs energy from the environment. The symbol N indicates the areas of the negative energy flows, in which the system releases energy to the environment. We set the parameter values to $\omega_{\mathrm{h}}=5$, $\omega_{\mathrm{c}}=1$, $\beta_{\mathrm{h}}=0.2$, $\beta_{\mathrm{c}}=1$, and hence $\omega_{\mathrm{h}}/\omega_{\mathrm{c}} = \beta_{\mathrm{c}}/\beta_{\mathrm{h}} = 1/5$.
• Figure 5: Operation of the Otto cycle depending on the ratio $g_\alpha/\omega_\alpha$ ($\alpha = \mathrm{h}, \mathrm{c}$) in the Gibbs-state limit cycle (GSLC). In the blue areas on the left side of each figure, the Otto cycle operates as a refrigerator. In the green areas in the middle, the Otto cycle operates as an engine. In the white areas, the Otto cycle cannot run as a thermal machine. We set the energy $\omega_{\mathrm{c}}$ and the inverse temperature $\beta_{\mathrm{c}}$ to unity, fix the inverse temperatures as $\beta_{\mathrm{c}}/\beta_{\mathrm{h}}=5$, and increase the energy level $\omega_{\mathrm{h}}$ from (a) to (c) as $\omega_{\mathrm{h}}/\omega_{\mathrm{c}}=2, 5, 6$.