  We study the entanglement between two coupled detectors, whose internal
degrees of freedom are modeled by harmonic oscillators, interacting with a
common quantum field, paying special attention to two less studied yet
important features: finite separation and direct coupling. Distance dependence
is essential in quantum teleportation and relativistic quantum information
considerations. The presence of a quantum field as the environment accords an
indirect interaction between the two oscillators at finite separation of a
non-Markovian nature which competes with the direct coupling between them. The
interplay between these two factors results in a rich variety of interesting
entanglement behaviors at late times. We show that the entanglement behavior
reported in prior work assuming no separation between the detectors can at best
be a transient effect at very short times, and claims that such behaviors
represent late time entanglement are misplaced. Entanglement between the
detectors with direct coupling enters in the consideration of macroscopic
quantum phenomena and other frontline issues. We find that with direct coupling
entanglement between the two detectors can sustain over a finite distance, in
contrast to the no-direct coupling case reported before, where entanglement can
not survive at separation more than a few inverse high frequency cutoff scales.
This work provides a solid platform necessary for further systematic
investigations into the entanglement behavior of continuous variable quantum
systems.
