Non-equilibrium Dynamics of Three-Level Absorption Refrigerator at Third-Order Liouvillian Exceptional Points

Abstract

We analyze the influence of Liouvillian exceptional points (LEPs) in the three-level quantum absorption refrigerator, putting emphasis on the non-equilibrium process before the convergence to the steady state. We search for the second-order and third-order LEPs in the system with two types of couplings. Focusing on the third-order LEPs, we analyze the damping of the system state in the long term analytically and numerically. In addition, we analyze the damping of heat currents and the influence of the non-equilibrium process in the heat extraction from the cold bath. Critical damping at LEPs of both the system state and the heat currents is achieved, implying the fastest convergence to the equilibrium system. During the non-equilibrium process, we find that much heat transfer from the cold bath to the hot bath with less energy cost of the work bath is achieved at the third-order LEP, leading to better performance of the refrigerator.

Figures (14)

• Figure 1: Schematic views of the three-level absorption refrigerator. (a) Energy levels and couplings of the internal three-level system. (b) Interaction between the internal three-level system and the external three heat baths.
• Figure 2: Schematic views of the three-level system with (a) $g_\textrm{w}$-coupling and (b) $g_\textrm{c}$-coupling.
• Figure 3: Possible Distribution of the five eigenvalues with all real-positive parameters $\{\gamma^\pm, \omega_\alpha, g_\alpha\}$. (a) and (b) are possible distributions without LEPs, (c) is the distribution with a second-order LEP, and (d) is the distribution with a third-order LEP. Note that the sequence of the eigenvalue indices may be different. In (a), there are various orderings of the four real eigenvalues, and $\lambda_1$ is not always the smallest one. In (b) and (c), $\lambda_1$ and $\lambda_2$ may be greater than $\textrm{Re}[\lambda_{3,4}]$ and $\lambda^{(2)}$. In (d), $\lambda_1$ may be greater than $\lambda_{(3)}$.
• Figure 4: Time-dependence of the ratio $\mathcal{R}_\textrm{s}(t)$ for (a) the $g_\textrm{w}$-coupling system and (b) the $g_\textrm{c}$-coupling system. We specify the energy levels as $\omega_\textrm{w}=\omega_\textrm{c}=1$ as the energy unit and set the dissipation rate $\gamma^+$ as in Eq. (\ref{['EP3Para2']}) for all states. Following the condition (\ref{['EP3Para1']}), we set the coupling strength $g_\alpha=(\sqrt{2}/2)\omega_\alpha$ at the third-order LEP. We choose the values of the coupling strength as $g_\alpha=1.314$ for the state near LEP ("near-LEP") and $g_\alpha=0.414$ for the state away from LEP ("away-LEP"). The parameters $\{\gamma^-,c_1,c_2,c_3,c_4\}$ are $\{7.159, 0, 0.125, 0.1,0\}$ for (a) the $g_\textrm{w}$-coupling system while $\{5.857, 0.127,0.1,0, 0\}$ for (b) the $g_\textrm{c}$-coupling system
• Figure 5: Time-dependence of the ratio $\mathcal{R}_\textrm{c}(t)$ for (a) the $g_\textrm{w}$-coupling system and (b) the $g_\textrm{c}$-coupling system. The other parameters $\{\gamma^-,c_1,c_2,c_3,c_4\}$ are $\{7.159, 0, 0.125, 0.1,0\}$ for (a) the $g_\textrm{w}$-coupling system while $\{5.857, 0.127,0.1,0, 0\}$ for (b) the $g_\textrm{c}$-coupling system.