  The characterization of excitations in disordered quantum systems is a
central issue in connection with glass physics and many-body localization.
Here, we show that quench spectroscopy of a disordered model, as realized from
its out-of-equilibrium dynamics following a global quench, allows us to fully
characterize the spectral properties of the disordered phases. In the
Bose-Hubbard model, a clear signature of gapless excitations in
momentum-resolved spectroscopy enables us to accurately locate the Mott
insulator to Bose glass transition, while the presence or absence of a
well-defined soundlike mode distinguishes the superfluid from the Bose glass
phase. Moreover, spatially-resolved spectroscopy provides local spectral
properties and allows us to extract the typical spacing of gapless regions,
giving a second independent way to uniquely identify all three phases. Our
findings have far-ranging implications for a variety of experimental platforms,
and offer a powerful and versatile probe of the low-energy phases of disordered
systems.
