  We use a Rydberg quantum simulator to demonstrate a new form of spectroscopy,
called quench spectroscopy, which probes the low-energy excitations of a
many-body system. We illustrate the method on a two-dimensional simulation of
the spin-1/2 dipolar XY model. Through microscopic measurements of the spatial
spin correlation dynamics following a quench, we extract the dispersion
relation of the elementary excitations for both ferro- and anti-ferromagnetic
couplings. We observe qualitatively different behaviors between the two cases
that result from the long-range nature of the interactions, and the frustration
inherent in the antiferromagnet. In particular, the ferromagnet exhibits
elementary excitations behaving as linear spin waves. In the anti-ferromagnet,
spin waves appear to decay, suggesting the presence of strong nonlinearities.
Our demonstration highlights the importance of power-law interactions on the
excitation spectrum of a many-body system.
