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This paper is concerned with translation invariant networks of linear quantumstochastic systems with nearest neighbour interaction mediated by boson fields.The systems are associated with sites of a one-dimensional chain or amultidimensional lattice and are governed by coupled linear quantum stochasticdifferential equations (QSDEs). Such interconnections of open quantum systemsare relevant, for example, to the phonon theory of crystalline solids, atomtrapping in optical lattices and quantum metamaterials. In order to represent alarge-scale open quantum harmonic oscillator, the coefficients of the coupledQSDEs must satisfy certain physical realizability conditions. These areestablished in the form of matrix algebraic equations for the parameters of anindividual building block of the network and its interaction with theneighbours and external fields. We also discuss the computation of mean squareperformance functionals with block Toeplitz weighting matrices for such systemsin the thermodynamic limit per site for unboundedly increasing fragments of thelattice.

Physical Realizability and Mean Square Performance of Translation Invariant Networks of Interacting Linear Quantum Stochastic Systems