Pure Gaussian states from quantum harmonic oscillator chains with a single local dissipative process

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.