  We present a theoretical description of a system of many spins strongly
coupled to a bosonic chain. We rely on the use of a spin-wave theory describing
the Gaussian fluctuations around the mean-field solution, and focus on
spin-boson chains arising as a generalization of the Dicke Hamiltonian. Our
model is motivated by experimental setups such as trapped ions, or atoms/qubits
coupled to cavity arrays. This situation corresponds to the cooperative
(E$\otimes$$\beta$) Jahn-Teller distortion studied in solid-state physics.
However, the ability to tune the parameters of the model in quantum optical
setups opens up a variety of novel intriguing situations. The main focus of
this paper is to review the spin-wave theoretical description of this problem
as well as to test the validity of mean-field theory. Our main result is that
deviations from mean-field effects are determined by the interplay between
magnetic order and mesoscopic cooperativity effects, being the latter strongly
size-dependent.
