Synthetic band-structure engineering in polariton crystals with non-Hermitian topological phases

Synthetic crystal lattices provide ideal environments for simulating and exploring the band structure of solid-state materials in clean and controlled experimental settings. Physical realisations have, so far, dominantly focused on implementing irreversible patterning of the system, or interference techniques such as optical lattices of cold atoms. Here, we realise reprogrammable synthetic band-structure engineering in an all optical exciton-polariton lattice. We demonstrate polariton condensation into excited states of linear one-dimensional lattices, periodic rings, dimerised non-trivial topological phases, and defect modes utilising malleable optically imprinted non-Hermitian potential landscapes. The stable excited nature of the condensate lattice with strong interactions between sites results in an actively tuneable non-Hermitian analogue of the Su-Schrieffer-Heeger system.