Quantum transport with coupled cavities on the Apollonian network

We study the dynamics of atomic and photonic excitations in the Jaynes-Cummings-Hubbard (JCH) model when each cavity occupies the nodes of an Apollonian network (AN). Such structure shows small-world and scale-free properties, and has been investigated within many physical models. By numerically diagonalizing the system Hamiltonian in the single-excitation subspace, we evaluate the time evolution of an initial state prepared as an even superposition of both atomic and photonic modes fully localized at a specific node. For the large hopping regime we provide a detailed description of the quantum transport properties of the AN and show that the complex interplay between the many cavity-cavity degrees of freedom induces both extended and localized states varying periodically. The excitation is most likely to be found in the initial node and its propagation strongly depends on the initial conditions. We also discuss the strong atom-cavity coupling regime, where atomic and photonic modes propagates identically, and the JCH regime, where the system is allowed to roam between atomic and photonic degrees of freedom, since the hopping parameter and the atom-cavity coupling strength are of the same order. In this regime, different cavities contribute either to the atomic or photonic component mostly, depending on the initial conditions.