Flat-band quantum geometry from embedded quench dynamics

Quantum geometry underlies a wide range of correlated phenomena in electronic systems, yet accessing the geometry of individual bands remains experimentally challenging. Advances in quantum simulators, such as ultracold atoms in optical lattices, have enabled local state preparation and site-resolved measurements, offering direct access to quantum dynamics beyond conventional transport probes. These developments naturally raise the question of whether band-resolved quantum geometry can be extracted from local occupation readouts. Here, we propose an on-site quench protocol to probe the flat-band quantum geometry via local occupation measurements in an atomic lattice. Our scheme dynamically isolates the flat-band sector by coupling a one-dimensional flat-band lattice to a two-dimensional flat-band bath. The bath selectively removes dispersive excitations, while compact localized states remain trapped due to their overlap with the flat-band modes of the bath. We show that the late-time occupation pattern directly encodes the integrated overlaps between flat-band states separated by the wavevector of the initial density pattern, thereby enabling access to integrated quantum distances between the states separated by the commensurate momenta δk = 2 πs/r, s,r in Z and to the integrated flat-band quantum metric. Our results thereby establish a direct connection between quantum geometry and the long-time memory of an initial spatial pattern in flat-band systems.