Quantum Entanglement under Non-Markovian Dynamics of Two Qubits Interacting with a common Electromagnetic Field

We study the non-Markovian dynamics of a pair of qubits made of two-level atoms separated in space with distance $r$ and interacting with one common electromagnetic field but not directly with each other. Our calculation makes a weak coupling assumption, but no Born or Markov approximation. We use the concurrence function as a measure of quantum entanglement between the two qubits. Two classes of states are studied in detail: the GHZ state and the Bell state. Comparison with earlier work on the same model but using the Born-Markov approximation shows similar behavior in the Bell states but qualitatively different behavior in the evolution of the GHZ state. Under the non-Markovian evolution, for an initial GHZ state we do not see sudden death of quantum entanglement and subsequent revivals, except when the qubits are sufficiently far apart. For an initial Bell state, our findings based on non-Markovian dynamics agrees with those obtained under the Born-Markov approximation. We provide explanations for such differences of behavior both between these two classes of states and between the predictions from the Markov and non-Markovian dynamics. We also studied the decoherence of this two-qubit system and find that the decoherence rate in the case of one qubit initially in an excited state does not change significantly with the qubits separation whereas it does for the case when one qubit is initially in the ground state. Furthermore, when the two qubits are close together, the coherence of the whole system is preserved longer than it does in the single qubit case or when the two qubits are far apart