Witnessing entanglement between two two-level atoms coupled to a leaky cavity via two-photon relaxation

We analyzed the exact Markovian and non-Markovian dynamics of two atoms, allowing for dipole-dipole and Ising-like interplays between them, coupled asymmetrically to a leaky cavity via two-photon relaxation. Beside analyzing the conditions to preserve the initial atoms entanglement trough relaxation into the steady state of the system, we establish a relationship between the stationary entanglement and quantum-memory-assisted entropic uncertainty relation (QMA-EUR). Then, we discuss the effects of atom-atom interaction and the asymmetry in the couplings between each atom and cavity on the dynamics of entanglement and QMA-EUR in the Markovian and non-Markovian regimes, and investigate the possibility of witnessing the entanglement by the lower bound of QMA-EUR. We show that, when atoms asymmetrically are coupled to a Markovian environment, regularly increasing the Ising interaction strength reduces the lower bound of the entropic uncertainty relation, prolongs the entanglement witness time, and effectively protect the entanglement region witnessed by the lower bound of QMA-EUR. What's more, without the dipole-dipole interaction, this entanglement is always witnessed during the interaction. Although, dipole-dipole interaction destroys the stationary entanglement, increases the entropic uncertainty, and shortens the entanglement witness time, we find a relatively strong dipole-dipole interaction suppresses the disentanglement and prevents shortening of the entanglement witness time. Our results indicate that, if the dipole-dipole interaction is also present, the entanglement and its witness are always favored by the Ising interaction and the symmetric or asymmetric coupling configuration.