Dynamics of the entanglement between two oscillators in the same environment

We provide a complete characterization of the evolution of entanglement between two resonant oscillators coupled to a common environment. We identify three phases with different qualitative long time behavior: There is a phase where entanglement undergoes a sudden death. Another phase (sudden death and revival) is characterized by an infinite sequence of events of sudden death and revival of entanglement. In the third phase (no sudden death) there is no sudden death of entanglement, which persists for a long time. The phase diagram is described and analytic expressions for the boundary between phases are obtained. These results are applicable to a large variety of non-Markovian environments. The case of nonresonant oscillators is also numerically investigated.