Quantum transport with coupled cavities on the Apollonian network

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.