Strong system-bath coupling induces negative differential thermal conductance and heat amplification in nonequilibrium two-qubit systems

Quantum heat transfer is analyzed in nonequilibrium two-qubits systems by applying the nonequilibrium polaron-transformed Redfield equation combined with full counting statistics. Steady-state heat currents with weak and strong qubit-bath couplings are clearly unified. Within the two-terminal setup, the negative differential thermal conductance is unraveled with strong qubit-bath coupling and finite qubit splitting energy. The partially strong spin-boson interaction is sufficient to show the negative differential thermal conductance. Based on the three-terminal setup, in which two qubits are asymmetrically coupled to three thermal baths, a giant heat amplification factor is observed with strong qubit-bath coupling. Moreover, the strong interaction of either the left or right spin-boson coupling is able to exhibit the apparent heat amplification effect.

Figure 1

Schematic illustration of quantum thermal transfer in two-qubit systems driven by (a) two thermal baths (square panels with temperatures $T_L$ and $T_R$) with $I$ the heat flow and (b) three thermal baths (square panels with temperatures $T_{L_h}, T_{L_c}$, and $T_R$) with $I_{L_{h\left(c\right)}}$ and $I_R$ heat currents. The circles describe two-level qubits with horizontal solid lines denoting the qubit energy levels, the wavelike curves show the interaction between thermal baths and qubits system, and the double-arrowed brown lines describe the interqubit coupling.

Figure 2

Comparison of steady-state heat currents by tuning qubit-bath coupling strength $\alpha$ from the NE-PTRE [blue (dark) and red (light) solid lines], the Redfield equation [blue (dark) and red (light) circles], and the nonequilibrium NIBA scheme [blue (dark) and red (light) squares], respectively. The energy bias is selected as $\varepsilon =0$ for the blue (dark) solid line, circles and squares, and $\varepsilon =1$ for the red (light) solid line, circles and squares. The other parameters are given by $\mathrm{\Delta }=1, U=0.1, {\omega }_c=5, T_L=1.5$, and $T_R=0.5$.

Figure 3

Normalized heat current $I/I_{\text{max}}\left(\alpha \right)$ by (a) tuning temperature difference $\mathrm{\Delta }T=T_L-T_R$ with typical qubit-bath coupling strength $\alpha$ and (b) both modulating $\mathrm{\Delta }T$ and $\alpha$ in a 3D view, with $I_{max}\left(\alpha \right)={max}_{\mathrm{\Delta }T}\left\{I\right\}$ for a given $\alpha$. The temperatures are given by $T_L=T_0+\mathrm{\Delta }T/2$ and $T_R=T_0-\mathrm{\Delta }T/2$ with $T_0=2$. The other parameters are $\varepsilon =1, \mathrm{\Delta }=1, U=0.1$, and ${\omega }_c=5$.

Figure 4

(a) Schematic illustration of the heat transfer process for one current component $4U{\kappa }_L^{24}{\kappa }_R^{43}{\kappa }_L^{31}{\kappa }_R^{12}/\mathcal{A}$ in Eq. (19) in the strong qubit-bath coupling limit with the states $|1\rangle =|\uparrow \uparrow \rangle , |2\rangle =|\uparrow \downarrow \rangle , |3\rangle =|\downarrow \uparrow \rangle$, and $|4\rangle =|\downarrow \downarrow \rangle$, the transition rate ${\kappa }_v^{ij}(v=L,R)$ in Eq. (20), and horizontal (red-light and blue-dark) arrows with wide and thin sizes representing different amplitudes of ${\kappa }_v^{ij}$; (b) the corresponding transition rates and (c) two-loop currents and $I_{\mathrm{NIBA}}$ in Eq. (19) as a function of $\mathrm{\Delta }T$ with $\alpha =5$. The other parameters are the same as in Fig. 3.

Figure 5

Normalized heat current $I/I_{\text{max}}\left({\alpha }_R\right)$ (a) by tuning temperature bias $\mathrm{\Delta }T$ with typical ${\alpha }_R$, and (b) by modulating both $\mathrm{\Delta }T$ and ${\alpha }_R$ in a 3D view, with $I_{\text{max}}\left({\alpha }_R\right)={max}_{\mathrm{\Delta }T}\left\{I\right\}$ at given ${\alpha }_L$ and ${\alpha }_R$. The left qubit-bath coupling strength is ${\alpha }_L=0.05$ and the other parameters are the same as in Fig. 3.

Figure 6

(a) Heat amplification factor ${\beta }_{L_h}$ with typical qubit-bath couplings and (b) steady-state currents with strong coupling ($\alpha =5$), by varying the temperature of the right bath $T_R$. The other parameters are given by $\varepsilon =1, \mathrm{\Delta }=1, U=0.1, {\omega }_c=5, T_{L_h}=2$, and $T_{L_c}=0.2$.

Figure 7

Heat amplification factor ${\beta }_{L_h}$ as a function of $T_R$ at (a) weak (${\alpha }_{L_h}={\alpha }_{L_c}=0.05$) and (b) strong (${\alpha }_{L_h}={\alpha }_{L_c}=3$) left qubit-bath couplings, with various right qubit-bath coupling strengths. The other parameters are given by $\varepsilon =1, \mathrm{\Delta }=1, U=0.1, {\omega }_c=5, T_{L_h}=2$, and $T_{L_c}=0.2$.

Figure 8

Dynamics of the real part of off-diagonal density matrix elements. The system parameters are given by $\varepsilon =1, \mathrm{\Delta }=1, U=0.1, \alpha =0.01, {\omega }_c=5, T_L=1.5$, and $T_R=0.5$.

Figure 9

Components of the heat current $J_{\mathrm{weak}}$ within the Redfield scheme in Eq. (A6). The bath temperatures are given by $T_L=T_0+\mathrm{\Delta }T/2$ and $T_R=T_0-\mathrm{\Delta }T/2$ with $T_0=2$. The other system parameters are given by $\varepsilon =1, \mathrm{\Delta }=1, U=0.1, \alpha =0.01$, and ${\omega }_c=5$.

Figure 10

Normalized heat current $I/I_{\text{max}}\left(\alpha \right)$ with various (a) qubit splitting energies $\varepsilon$ and (b) qubit-qubit repulsion strengths $U$, loop currents, and $I_{\text{strong}}$ with (c) $\varepsilon =0$ and (d) $U=0.8$, by tuning temperature bias $\mathrm{\Delta }T$ with strong qubit-bath coupling $\alpha =5$. The other parameters are the same as in Fig. 3.

Figure 11

Heat current and components in Eq. (C14) within the three-terminal setup with strong qubit-bath coupling ($\alpha =5$) as a function of $T_R$. The other parameters are $\varepsilon =1, \mathrm{\Delta }=1, U=0.1, {\omega }_c=5, T_{L_h}=2$, and $T_{L_c}=0.2$.