Distinguishing trivial and topological quadrupolar insulators by Wannier-Stark ladders

I study theoretically quadrupolar topological insulators under applied static electric field rotated along the crystal axis. I demonstrate that the energy spectrum of this structure is a Wannier-Stark ladder that naturally visualizes the quantization of nested Wilson loops for the bulk bands. This enables a direct distinction between the topological phase, possessing localized corner states, and the trivial phase, lacking the corner states. The crossover between the topological and trivial cases takes place via the nonadiabatic regime. These results may find applications in the characterization of rapidly emerging higher-order topological phases of light and matter.