  We study the preparation of entangled pure Gaussian states via reservoir
engineering. In particular, we consider a chain consisting of $(2\aleph+1)$
quantum harmonic oscillators where the central oscillator of the chain is
coupled to a single reservoir. We then completely parametrize the class of
$(2\aleph+1)$-mode pure Gaussian states that can be prepared by this type of
quantum harmonic oscillator chain. This parametrization allows us to determine
the steady-state entanglement properties of such quantum harmonic oscillator
chains.
