  We consider an array of coupled cavities with staggered inter-cavity
couplings, where each cavity mode interacts with an atom. In contrast to
large-size arrays with uniform-hopping rates where the atomic dynamics is known
to be frozen in the strong-hopping regime, we show that resonant atom-field
dynamics with significant energy exchange can occur in the case of staggered
hopping rates even in the thermodynamic limit. This effect arises from the
joint emergence of an energy gap in the free photonic dispersion relation and a
discrete frequency at the gap's center. The latter corresponds to a bound
normal mode stemming solely from the finiteness of the array length. Depending
on which cavity is excited, either the atomic dynamics is frozen or a
Jaynes-Cummings-like energy exchange is triggered between the bound photonic
mode and its atomic analogue. As these phenomena are effective with any number
of cavities, they are prone to be experimentally observed even in small-size
arrays.
