  Quenches are now routinely used in synthetic quantum systems to study a
variety of fundamental effects, including ergodicity breaking, light-cone-like
spreading of information, and dynamical phase transitions. It was shown
recently that the dynamics of equal-time correlators may be related to
ground-state phase transitions and some properties of the system excitations.
Here, we show that the full low-lying excitation spectrum of a generic
many-body quantum system can be extracted from the after-quench dynamics of
equal-time correlators. We demonstrate it for a variety of one-dimensional
lattice models amenable to exact numerical calculations, including Bose and
spin models, with short- or long-range interactions. The approach also applies
to higher dimensions, correlated fermions, and continuous models. We argue that
it provides an alternative approach to standard pump-probe spectroscopic
methods and discuss its advantages.
