  We study the non-equilibrium 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 derived a
non-Markovian master equation for the evolution of the reduced density matrix
of the two-qubit system after integrating out the electromagnetic field modes.
It contains a Markovian part with a Lindblad type operator and a nonMarkovian
contribution, the physics of which is the main focus of this study. We use the
concurrence function as a measure of quantum entanglement between the two
qubits. Two classes of states are studied in detail: Class A is a one parameter
family of states which are the superposition of the highest energy and Class B
states are linear combinations of the symmetric and the antisymmetric Bell
states. We obtain similar behavior for the Bell states as in earlier results
derived by using the Born-Markov approximation. However, in the Class A states
the behavior is qualitatively different: under the non-Markovian evolution we
do not see sudden death of quantum entanglement and subsequent revivals, except
when the qubits are sufficiently far apart. 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 study 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.
