Dephasing of two interacting qubits under the influence of thermal reservoirs

A phase relaxation process of interacting two qubits is studied by means of the quantum master equation, where each qubit is influenced by an independent thermal reservoir. The damping operator of the quantum master equation consists of two parts; one includes the effects of the interaction between the qubits and the other does not. It is shown that the interaction significantly affects the phase relaxation process. The decoherence of entanglement is investigated for the two qubits initially prepared in the $X$ state.

Figure 1

The schematic representation of the interacting qubits under the influence of the individual thermal reservoirs, where ${\widehat{S}}_{A,B}^z$ and ${\widehat{S}}_{A,B}^{\pm }$ are spin-1/2 operators and ${\widehat{R}}_{A,B}$ is a Hermitian operator of the thermal reservoir. The interaction between the qubits affects the relaxation process.

Figure 2

(Color online) The time evolution of the concurrence $C\left(t\right)$ of the two qubits initially prepared in $|\Psi ⟩$, where we set (a-1) $\alpha =0.5$, $\varphi =0.0$, (b-1) $\alpha =0.5$, $\varphi =\pi /4$, (c-1) $\theta =0.0$, $\varphi =\pi /4$, and (d-1) $\theta =\pi /2$, $\varphi =\pi /4$. The figures (a-2), (b-2), (c-2), and (d-2) show the concurrence when the interaction effects on the relaxation are ignored, where the values of the parameters are the same as those for (a-1), (b-1), (c-1), and (d-1). In all the figures, we set $T_{\text{ph}}/{\tau }_c=2.0$.

Figure 3

(Color online) The time evolution of the concurrence $C\left(t\right)$, where we set (a-1) $T_{\text{ph}}/{\tau }_c=5.0$, $\varphi =\pi /4$, $a=0.2$ and (b-1) $T_{\text{ph}}/{\tau }_c=5.0$, $\alpha =0.8$, $a=0.2$. The figures (a-2) and (b-2) show the concurrence when the interaction effects on the relaxation are ignored, where the values of the parameters are the same as those for (a-1) and (b-1).

Figure 4

(Color online) The time evolution of the concurrence $C\left(t\right)$, where we set (a-1) $T_{\text{ph}}/{\tau }_c=5.0$, $\alpha =0.8$, (b-1) $T_{\text{ph}}/{\tau }_c=5.0$, $r=0.8$, and (c-1) $T_{\text{ph}}/{\tau }_c=5.0$, $r=0.6$. The figures (a-2), (b-2), and (c-2) show the concurrence when the interaction effects on the relaxation are ignored, where the values of the parameters are the same as those for (a-1), (b-1), and (c-1).