Finite Su-Schrieffer-Heeger chains coupled to a two-level emitter: Hybridization of edge and emitter states

The Hamiltonian for the one-dimensional Su-Schrieffer-Heeger (SSH) chain is one of the simplest Hamiltonians that supports topological states. This work considers between one and three finite SSH chains with open boundary conditions that either share a lattice site (or cavity), which---in turn---is coupled to a two-level emitter, or are coupled to the same two-level emitter. We investigate the system properties as functions of the emitter-cavity coupling strength $g$ and the detuning between the emitter energy and the center of the band gap. It is found that the energy scale introduced by the edge states that are supported by the uncoupled finite SSH chains leads to a $g$-dependent hybridization of the emitter and edge states that is unique to finite-chain systems. A highly accurate analytical three-state model that captures the band-gap physics of $k$-chain $(k\ensuremath{\ge}1)$ systems is developed. To quantify the robustness of the topological system characteristics, the inverse participation ratio for the cavity-shared and emitter-shared systems consisting of $k$ chains is analyzed as a function of the on-site disorder strength. The $g$-dependent hybridization of the emitter and uncoupled edge states can be probed dynamically.