Generation of Long-distance Many-body Entangled States in Atoms Coupled to Waveguides.

Generating entangled many-body states between macroscopically separated atoms is an essential resource for distributed quantum sensing and quantum communication applications. In this work, we develop and theoretically demonstrate a probabilistic protocol to herald many-body atomic entangled states in an ordered array of multilevel atoms coupled to a waveguide. Initializing the atomic array in an uncorrelated state, with one excited 'emitter atom' in front of the array, we find that the subsequent collective dynamics of the atomic array and the photon emitted by the 'emitter atom’ are entangled. We thus propose an optimal configuration for a photodetection measurement on the scattered field to herald a desired entangled state of the array atoms, specifically a Greenberger-Horne-Zeilinger (GHZ) state. We demonstrate that successive iterations of the protocol increase the probability of heralding the target state, limited by the probability of photodetection. Our protocol incorporates large separation distances between atoms in the non-Markovian regime.