We study the steady state of two coupled two-level atoms interacting with a non-equilibrium environment that consists of two heat baths at different temperatures. Specifically, we analyze four cases with respect to the configuration about the interactions between atoms and heat baths. Using secular approximation, the conventional master equation usually neglects steady-state coherence, even when the system is coupled with a non-equilibrium environment. When employing the master equation with no secular approximation, we find that the system coherence in our model, denoted by the off-diagonal terms in the reduced density matrix spanned by the eigenvectors of the system Hamiltonian, would survive after a long-time decoherence evolution. The absolute value of residual coherence in the system relies on different configurations of interaction channels between the system and the heat baths. We find that a large steady quantum coherence term can be achieved when the two atoms are resonant. The absolute value of quantum coherence decreases in the presence of additional atom-bath interaction channels. Our work sheds new light on the mechanism of steady-state coherence in microscopic quantum systems in non-equilibrium environments.
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Figure 1
(Color online) The diagram sketch of our model with two coupled atoms interacting with a non-equilibrium environment consisted of a hotter bath-$b$ and a colder bath-$a$. The coupling coefficients $c_{i\alpha}$'s, $i=1,2$, $\alpha=a,b$, determine the configuration of interaction channels.
Figure 2
(Color online) The absolute value of $\rho_{32}$ in configuration of Case A vs. the temperature difference $\Delta_{T}/\gamma$ between the two baths (a) and the temperature of the colder bath-$a$ $T_{a}/\gamma$ (b), under various detuning between the two atoms $\Delta/\gamma$. We fix $T_{a}/\gamma=10$ in (a) and $\Delta_{T}/\gamma=50$ in (b). The average frequency of atoms and the coupling strength between atoms are set as $\Omega/\gamma=30$ and $\xi/\gamma=2$, respectively.
Figure 3
(Color online) The phase diagram of $|\rho_{32}|$ for Case A in space of the temperature of bath-$a$ $T_{a}/\gamma$ and the temperature difference $\Delta_{T}/\gamma$. Here we consider the resonance case for the two atoms that $\Delta=0$. The average frequency of atoms and the coupling strength between atoms are set as $\Omega/\gamma=30$ and $\xi/\gamma=2$, respectively.
Figure 4
(Color online) The phase diagram of $|\rho_{32}|$ for Case B in space of the temperature of bath-$a$ $T_{a}/\gamma$ and the temperature difference $\Delta_{T}/\gamma$. We consider the resonant condition with $\Delta=0$ and other parameters are set as the same as
Figure 5
(Color online) The phase diagram of $|\rho_{32}|$ for Case C in space of the temperature of bath-$a$ $T_{a}/\gamma$ and the temperature difference $\Delta_{T}/\gamma$. We consider the resonant condition with $\Delta=0$ and other parameters are set as the same as
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