  We study the dynamics of single photonic and atomic excitations in the
Jaynes-Cummings-Hubbard (JCH) model where the cavities are arranged in an
Apollonian network (AN). The existence of a gapped field normal frequency
spectrum along with strongly localized eigenstates on the AN highlights many of
the features provided by the model. By numerically diagonalizing the JCH
Hamiltonian in the single excitation subspace, we evaluate the time evolution
of fully localized initial states, for many energy regimes. We provide a
detailed description of the photonic quantum walk on the AN and also address
how an effective Jaynes-Cummings interaction can be achieved at the strong
hopping regime. When the hopping rate and the atom-field coupling strength is
of the same order, the excitation is relatively allowed to roam between atomic
and photonic degrees of freedom as it propagates. However, different cavities
will contribute mostly to one of these components, depending on the detuning
and initial conditions, in contrast to the strong atom-field coupling regime,
where atomic and photonic modes propagate identically.
