Topological photonic states in one-dimensional dimerized ultracold atomic chains

We study the topological optical states in one-dimensional (1D) dimerized ultracold atomic chains, as an extension of the Su-Schrieffer-Heeger (SSH) model. By taking the fully retarded near-field and far-field dipole-dipole interactions into account, we describe the system by an effective non-Hermitian Hamiltonian, vastly different from the Hermitian Hamiltonian of the conventional SSH model. We analytically calculate the complex bandstructures for infinitely long chains, and show that the topological invariant, i.e., the complex Zak phase, is still quantized and becomes nontrivial when the dimerization parameter $\beta>0.5$, despite the broken chiral symmetry and non-Hermiticity. We have verified the validity of the bulk-boundary correspondence for this non-Hermitian system by further analyzing the eigenstate distributions along with their inverse participation ratios (IPRs) for finite chains, where topologically protected edge states are unambiguously identified. We also reveal that such topological edge states are robust under symmetry-breaking disorders. For transverse eigenstates, we further discover the increase of localization length of topological edge states with the increase of lattice period due to the presence of strong far-field dipole-dipole interactions. Moreover, the ultra-strong scattering cross section and ultra-narrow linewidth of a single cold atom allow us to observe in more detail about topological states than in conventional systems, such as the frequency shift with respect to the single-atom resonance and the largely tunable bandgap. We envisage these topological photonic states can provide an efficient interface between light and matter.