Ultracold atom arrays in optical lattices emerge as an excellent playground for the integration of topological photonics and quantum optics. Here, we study high-order topological quantum optics in an ultracold atom metasurface intended to mimic the two-dimensional Su-Schrieffer-Heeger model. We find the existence of long-range interactions beyond nearest-neighbor ones leads to isolated corner states in the band gap, and show a corner atom can be addressed by a laser drive far away from it via these nontrivial states. We demonstrate the Purcell factor can be used as a powerful tool to examine the existence of topological edge and corner states. We predict topological edge states can mediate strong coherent interactions between two remote impurity quantum emitters while suppressing dissipative losses thanks to the higher-order topology, generating robust and long-lived quantum entanglement, without the need for additional photonic structures.
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