Dynamical phases for the evolution of the entanglement between two oscillators coupled to the same environment

We study the dynamics of the entanglement between two oscillators that are initially prepared in a general two-mode Gaussian state and evolve while coupled to the same environment. In a previous paper, we showed that there are three qualitatively different dynamical phases for the entanglement in the long-time limit: sudden death, sudden death and revival, and no sudden death [Paz and Roncaglia, Phys. Rev. Lett. 100, 220401 (2008)]. Here we generalize and extend those results along several directions: We analyze the fate of entanglement for an environment with a general spectral density providing a complete characterization of the evolution for Ohmic, sub-Ohmic, and super-Ohmic environments. We consider two different models for the interaction between the system and the environment (one where the coupling is through position and another where the coupling is symmetric in position and momentum). Finally, we show that for nonresonant oscillators, the final entanglement is independent of the initial state and may be nonzero at very low temperatures.