  This paper is concerned with translation invariant networks of linear quantum
stochastic systems with nearest neighbour interaction mediated by boson fields.
The systems are associated with sites of a one-dimensional chain or a
multidimensional lattice and are governed by coupled linear quantum stochastic
differential equations (QSDEs). Such interconnections of open quantum systems
are relevant, for example, to the phonon theory of crystalline solids, atom
trapping in optical lattices and quantum metamaterials. In order to represent a
large-scale open quantum harmonic oscillator, the coefficients of the coupled
QSDEs must satisfy certain physical realizability conditions. These are
established in the form of matrix algebraic equations for the parameters of an
individual building block of the network and its interaction with the
neighbours and external fields. We also discuss the computation of mean square
performance functionals with block Toeplitz weighting matrices for such systems
in the thermodynamic limit per site for unboundedly increasing fragments of the
lattice.
