  This paper is concerned with networks of identical linear quantum stochastic
systems which interact with each other and external bosonic fields in a
translation invariant fashion. The systems are associated with sites of a
multidimensional lattice and are governed by coupled linear quantum stochastic
differential equations (QSDEs). The block Toeplitz coefficients of these QSDEs
are specified by the energy and coupling matrices which quantify the
Hamiltonian and coupling operators for the component systems. We discuss the
invariant Gaussian quantum state of the network when it satisfies a stability
condition and is driven by statistically independent vacuum fields. A
quadratic-exponential functional (QEF) is considered as a risk-sensitive
performance criterion for a finite fragment of the network over a bounded time
interval. This functional involves a quadratic function of dynamic variables of
the component systems with a block Toeplitz weighting matrix. Assuming the
invariant state, we study the spatio-temporal asymptotic rate of the QEF per
unit time and per lattice site in the thermodynamic limit of unboundedly
growing time horizons and fragments of the lattice. A spatio-temporal
frequency-domain formula is obtained for the QEF rate in terms of two spectral
functions associated with the real and imaginary parts of the invariant quantum
covariance kernel of the network variables. A homotopy method and asymptotic
expansions for evaluating the QEF rate are also discussed.

Quantum Physics;Systems and Control;Systems and Control;Optimization and Control