We investigate the unitary evolution following a quantum quench in quantum spin models possessing a (nearly) flat band in the linear excitation spectrum. Inspired by the perspective offered by ensembles of individually trapped Rydberg atoms, we focus on the paradigmatic trasverse-field Ising model on two dimensional lattices featuring a flat band as a result of destructive interference effects (Lieb and Kagome lattice); or a nearly flat band due to a strong energy mismatch among sublattices (triangular lattice). Making use of linear spin-wave theory, we show that quantum quenches, equipped with single-spin imaging, can directly reveal the spatially localized nature of the dispersionless excitations, and their slow propagation or lack of propagation altogether. Moreover we show that Fourier analysis applied to the post-quench time evolution of wavevector-dependent quantities allows for the spectroscopic reconstruction of the flat bands. Our results pave the way for future experiments with Rydberg quantum simulators, which can extend our linear spin-wave study to the fully nonlinear regime, characterized by the appearance of dense, strongly interacting gases of dispersionless excitations.
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