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This paper is concerned with networks of identical linear quantum stochasticsystems which interact with each other and external bosonic fields in atranslation invariant fashion. The systems are associated with sites of amultidimensional lattice and are governed by coupled linear quantum stochasticdifferential equations (QSDEs). The block Toeplitz coefficients of these QSDEsare specified by the energy and coupling matrices which quantify theHamiltonian and coupling operators for the component systems. We discuss theinvariant Gaussian quantum state of the network when it satisfies a stabilitycondition and is driven by statistically independent vacuum fields. Aquadratic-exponential functional (QEF) is considered as a risk-sensitiveperformance criterion for a finite fragment of the network over a bounded timeinterval. This functional involves a quadratic function of dynamic variables ofthe component systems with a block Toeplitz weighting matrix. Assuming theinvariant state, we study the spatio-temporal asymptotic rate of the QEF perunit time and per lattice site in the thermodynamic limit of unboundedlygrowing time horizons and fragments of the lattice. A spatio-temporalfrequency-domain formula is obtained for the QEF rate in terms of two spectralfunctions associated with the real and imaginary parts of the invariant quantumcovariance kernel of the network variables. A homotopy method and asymptoticexpansions for evaluating the QEF rate are also discussed.

Infinite-horizon risk-sensitive performance criteria for translation invariant networks of linear quantum stochastic systems