Spontaneously generated atomic entanglement in free space: reinforced by incoherent pumping

We study spontaneously generated entanglement (SGE) between two identical multilevel atoms in free space via vacuum-induced radiative coupling. We show that the SGE in two-atom systems may initially increase with time but eventually vanishes in the time scale determined by the excited state lifetime and radiative coupling strength between the two atoms. We demonstrate that a steady-state SGE can be established by incoherently pumping the excited states of the two-atom system. We have shown that an appropriate rate of incoherent pump can help producing optimal SGE. The multilevel systems offer us more chanel to establish entanglement. The system under consideration could be realized in a tight trap or atoms/ions doped in a solid substrate.