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Traditional spectroscopy, by its very nature, characterizes properties of physical systems in the momentum and frequency domains. The most interesting and potentially practically useful quantum many-body effects however emerge from the deep composition of local, short-time correlations. Here, using inelastic neutron scattering and methods of integrability, we experimentally observe and theoretically describe a local, coherent, long-lived, quasiperiodically oscillating magnetic state emerging out of the distillation of propagating excitations following a local quantum quench in a Heisenberg antiferromagnetic chain. This "quantum wake" displays similarities to Floquet states, discrete time crystals and nonlinear Luttinger liquids.