  Layer-by-layer material engineering has enabled exotic quantum phenomena such
as interfacial superconductivity and the quantum anomalous Hall effect.
Meanwhile, deciphering electronic states layer-by-layer remains a fundamental
scientific challenge. This is exemplified by the difficulty in understanding
the layer origins of topological electronic states in magnetic topological
insulators, which is key to understanding and controlling topological quantum
phases. Here, we report a layer-encoded frequency-domain ARPES experiment on a
magnetic topological insulator (MnBi2Te4)(Bi2Te3) to characterize the layer
origins of electronic states. Infrared laser excitations launch coherent
lattice vibrations with the layer index encoded by the vibration frequency;
photoemission spectroscopy tracks the electron dynamics, where the layer
information is decoded in the frequency domain. This layer-frequency
correspondence reveals a surprising wavefunction relocation of the topological
surface state from the top magnetic layer into the buried second layer,
reconciling the controversy over the vanishing broken-symmetry energy gap in
(MnBi2Te4)(Bi2Te3) and its related compounds. The layer-frequency
correspondence can be harnessed to disentangle electronic states layer-by-layer
in a broad class of van der Waals superlattices.
