Steady atomic entanglement with different quantum feedbacks

We consider a model of two driven and collectively damped atoms with different quantum feedbacks. We find that the steady entanglement depends on the initial condition in the case of symmetric feedback but is independent of the initial condition in the case of nonsymmetric feedback. We also find that in the process of the transition from the symmetric feedback to the nonsymmetric feedback, the entanglement between the two atoms is discontinuous. Without the feedback (symmetric case), in the symmetric subspace the steady entanglement between the two atoms will be very small; then we apply a feedback just to one of the atoms (nonsymmetric case), the steady entanglement will increase greatly, which can reach C 0.82. It is very interesting that if we withdraw the feedback, the strong steady entanglement will remain, i.e., the whole process is irreversible, and reminds us of the hysteresis phenomena.