  The spatial localization of quantum states plays a central role in
condensed-matter phenomena, ranging from many-body localization to topological
matter. Building on the dissipation-fluctuation theorem, we propose that the
localization properties of a quantum-engineered system can be probed by
spectroscopy, namely, by measuring its excitation rate upon a periodic drive.
We apply this method to various examples that are of direct experimental
relevance in ultracold atomic gases, including Anderson localization,
topological edge modes, and interacting particles in a harmonic trap. Moreover,
inspired by a relation between quantum fluctuations and the quantum metric, we
describe how our scheme can be generalized in view of extracting the full
quantum-geometric tensor of many-body systems. Our approach opens an avenue for
probing localization, as well as quantum fluctuations, geometry and
entanglement, in synthetic quantum matter.
