  We study the dynamics of an one dimensional array of Jaynes-Cummings-Hubbard
system of arbitrary number of coupled cavities, each containing a two level
atom that interacts with a field mode. In particular, we consider propagation
of a single excitation quantum state for two different couplings of the
photonic modes of the adjacent cavities, namely, a translation invariant closed
chain of uniformly coupled cavities, and also a linear chain with nonuniform
parabolically varying intercavity coupling where the interaction Hamiltonian is
associated with the Jacobi matrix of the Krawtchouk polynomials. Using a
description via the delocalized atomic and field modes we observe that for a
large detuning of these two degrees of freedom atomic excitations propagate
without populating the field modes, and vice versa. For the near-resonance
scenario between these modes the atomic excitations, say, while propagating mix
with the photonic states. In the context of the parabolic coupling between
photons of adjacent cavities an arbitrary element of the time-dependent
correlation function between two arbitrary cavities may be expressed in closed
form for dominant values of the detuning parameter, when an exact transmission
of the quantum state at pre-specified times is realized.
