The evolution of the entanglement between oscillators that interact with the
same environment displays highly non-trivial behavior in the long time regime.
When the oscillators only interact through the environment, three dynamical
phases were identified and a simple phase diagram characterizing them was
presented. Here we generalize those results to the cases where the oscillators
are directly coupled and we show how a degree of mixidness can affect the final
entanglement. In both cases, entanglement dynamics is fully characterized by
three phases (SD: sudden death, NSD: no-sudden death and SDR: sudden death and
revivals) which cover a phase diagram that is a simple variant of the
previously introduced one. We present results when the oscillators are coupled
to the environment through their position and also for the case where the
coupling is symmetric in position and momentum (as obtained in the RWA). As a
bonus, in the last case we present a very simple derivation of an exact master
equation valid for arbitrary temperatures of the environment.
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