  The elementary excitations of quantum spin systems have generally the nature
of weakly interacting bosonic quasi-particles, generated by local operators
acting on the ground state. Nonetheless in one spatial dimension the nature of
the quasiparticles can change radically, since many relevant one-dimensional
$S=1/2$ Hamiltonians can be exactly mapped onto models of spinless fermions
with local hopping and interactions. Due to the non-local nature of the
spin-to-fermion mapping, observing directly the fermionic quasiparticle
excitations is impossible using local probes, which are at the basis of all the
forms of spectroscopy (such as neutron scattering) traditionally available in
condensed matter physics. Here we show theoretically that \emph{quench
spectroscopy} for synthetic quantum matter -- which probes the excitation
spectrum of a system by monitoring the nonequilibrium dynamics of its
correlation functions -- can reconstruct accurately the dispersion relation of
fermionic quasiparticles in spin chains. This possibility relies on the ability
of quantum simulation experiments to measure non-local spin-spin correlation
functions, corresponding to elementary fermionic correlation functions. Our
analysis is based on new exact results for the quench dynamics of quantum spin
chains; and it opens the path to probe arbitrary quasiparticle excitations in
synthetic quantum matter.
