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Non-equilibrium entangled steady state of two independent two-level systems

We determine and study the steady state of two independent two-level systems weakly coupled to a stationary non-equilibrium environment. Whereas this bipartite state is necessarily uncorrelated if the splitting energies of the two-level systems are different from each other, it can be entangled if they are equal. For identical two-level systems interacting with two bosonic heat baths at different temperatures, we discuss the influence of the baths temperatures and coupling parameters on their entanglement. Geometric properties, such as the baths dimensionalities and the distance between the two-level systems, are relevant. A regime is found where the steady state is a statistical mixture of the product ground state and of the entangled singlet state with respective weights 2/3 and 1/3.