Nonequilibrium thermal entanglement in a three-qubit $XX$ model

Making use of the master equation and effective Hamiltonian approach, we investigate the steady-state entanglement in a three-qubit $XX$ model. Both symmetric and nonsymmetric qubit-qubit couplings are considered. The system (the three qubits) is coupled to two bosonic baths at different temperatures. We calculate the steady state by the effective Hamiltonian approach and discuss the dependence of the steady-state entanglement on the temperatures and couplings. The results show that for symmetric qubit-qubit couplings, the entanglements between the nearest neighbors are equal, independent of the temperatures of the two baths. The maximum of the entanglement arrives at $T_L=T_R$. For nonsymmetric qubit-qubit couplings, however, the situation is totally different. The baths at different temperatures would benefit the entanglement and the entanglements between the nearest neighbors are no longer equal. By examining the probability distribution of each eigenstate in the steady state, we present an explanation for these observations. These results suggest that the steady entanglement can be controlled by the temperature of the two baths.

Figure 1

(Color online) A schematic representation of our model. The two-level systems are connected to two bosonic baths held at different temperatures, $T_L$ and $T_R$, respectively.

Figure 2

(Color online) Steady-state entanglements $C_{12}$ and $C_{13}$ as functions of the bath temperatures $T_L$ and $T_R$. Here, $\epsilon =1$ and $J_1=J_2=1$.

Figure 3

(Color online) Steady-state entanglements $C_{12}$ and $C_{13}$ as functions of the bath temperatures $T_L$ and $T_R$. Here, $\epsilon =3$ and $J_1=J_2=1$.

Figure 4

(Color online) (Left column) The steady state entanglement as functions of the mean bath temperature $T_M=\left(T_L+T_R\right)/2$ and the temperature difference $\Delta T=T_L-T_R$ in nonsymmetrical case. Right column shows the corresponding concurrences change with $\Delta T$ in the case of $T_M=0.2$ (blue solid), $T_M=0.4$ (green dashed), $T_M=0.6$ (red dotted), $T_M=0.8$ (cyan dash-dotted), and $T_M=1$ (pink circle). Here, $\epsilon =3$, $J_1=0.5$, and $J_2=2.5$.

Figure 5

(Color online) The probability distribution for eigenstates $|s_i⟩$, $i=3,4,\dots ,8$ in the steady state as a function of the temperature difference $\Delta T$. In top-left figure, we set $\epsilon =1$ and $J_1=J_2=1$, while in top-right figure, we choose $\epsilon =3$, $J_1=0.5$, and $J_2=2.5$. In both two figures, the mean bath temperature $T_M$ is fixed with $T_M=0.4$. In bottom figure, we plot the probability distribution for $|s_6⟩$ with different mean bath temperature. The mean bath temperatures correspond to the right column of Fig. 4.

Figure 6

(Color online) Steady-state entanglement as functions of the coupling strengths $J_1$ and $J_2$ in the case $\epsilon =3$, $T_M=0.5$, and $\Delta T=-0.4$.