Lindbladian approach for many-qubit thermal machines: enhancing the performance with geometric heat pumping by entanglement

TL;DR

The paper develops a thermodynamically consistent Lindblad framework for slowly driven many-qubit quantum thermal machines, separating geometric pumping from dissipative dissipation via Berry curvature and a parameter-space metric. By performing a systematic slow-driving expansion up to second order, it derives explicit expressions for work, heat currents, and entropy production, and identifies a geometric bound on pumped heat for non-interacting qubits, which can be surpassed through qubit-qubit interactions and asymmetric bath couplings. Numerical results for two interacting qubits reveal that entanglement and coupling asymmetries redistribute dissipation and can enhance pumped heat beyond the non-interacting Landauer-like bound, though effects depend on the driving protocol. The work establishes a general platform for dissipation, pumping, and performance optimization in driven quantum devices operating as heat engines, with implications for quantum thermodynamics and design of efficient nanoscale energy converters.

Abstract

We present a detailed analysis of slowly driven quantum thermal machines based on interacting qubits within the framework of the Lindblad master equation. By implementing a systematic expansion in the driving rate, we derive explicit expressions for the rate of work of the driving forces, the heat currents exchanged with the reservoirs, and the entropy production up to second order, ensuring full thermodynamic consistency in the linear-response regime. The formalism naturally separates geometric and dissipative contributions, identified by a Berry curvature and a metric in parameter space, respectively. Analytical results show that the geometric heat pumped per cycle is bounded by $k_B T N_q \ln 2$ for $N_q$ non-interacting qubits, in direct analogy with the Landauer limit for entropy change. This bound can be surpassed when qubit interactions and asymmetric couplings to the baths are introduced. Numerical results for the interacting two-qubit system reveal a non-trivial role of the interaction between qubits and the coupling between the qubits and the baths in the behavior of the dissipated power. The approach provides a general platform for studying dissipation, pumping, and performance optimization in driven quantum devices operating as heat engines.

Figures (6)

• Figure 1: Benchmark of the entropy--energy balance for the two-qubit system coupled to two thermal reservoirs $L$ and $R$ at equal temperature. The external fields follow the driving protocol of Eq. (\ref{['proto']}) with $B_0 = k_B T$, as illustrated in Fig. \ref{['fig:J0']}. Panel (a): the time-dependent first-order heat currents entering the system from reservoirs $L$ and $R$ (blue and orange lines) sum to the instantaneous rate of change of the system entropy (green line) (see Eq. \ref{['entrof']}). Panel (b): the sum of the second-order heat currents (blue) and the driving power (orange) reproduces the second-order entropy variation (green) (see Eq. \ref{['entro1']}). For the numerical calculations we take $J=0$ and $\eta=1.2$ in \ref{['ham_simplif']}, contact asymmetry $b=2$ in Eq. (\ref{['contacts']}), and reservoir parameters $g_L=g_R=0.002\,k_B T$, $\omega_C=120\,k_B T$ in Eq. (\ref{['gammas']}).
• Figure 2: Rotor field in the parameter space $\bm X=(B_x,B_z)$ for different configurations of the two-qubit system. The closed paths $C$ indicate the driving cycles for which the corresponding results are presented in the following figures. Panel (a) shows the non-interacting case, while panel (b) illustrates the case with inter-qubit coupling $J = 2\,k_B T$. All other parameters are the same as in Fig. \ref{['fig:consistencias']}.
• Figure 3: Maximum eigenvalue of the symmetric parts of $-\underline{\Omega}_L$, $-\underline{\Omega}_R$, and $\underline{\Lambda}$ in the parameter space $(B_x,B_z)$. Panels (a) and (b) show the asymmetric distribution of dissipation between reservoirs due to the coupling asymmetry, while $\underline{\Lambda}$ displays a symmetric pattern. System parameters as in Fig. \ref{['fig:consistencias']} with $J=0$.
• Figure 4: Maximum eigenvalue of the symmetric part of matrix $\underline{\Lambda}$ for interacting qubits with $J = 1\; k_B T$ (a) and $J = 2\; k_B T$ (b). The interaction modifies both the magnitude and distribution of dissipation compared to the non-interacting case. Other parameters as in Fig. \ref{['fig:consistencias']}.
• Figure 5: Figures of merit in circles defined by the protocol Eq. (\ref{['proto1']}). The system and coupling parameters are the same as for Figure \ref{['fig:consistencias']}, with different values of $J$.