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Description
Ultralight bosons around spinning black holes can form macroscopic gravitational atoms through superradiance. We show that attractive self-interactions can turn such systems into periodically varying lenses for gravitational waves. During the long-lived 211–322 quasi-equilibrium, coherent interference between the dominant 211 cloud and the self-interaction-populated 322 level produces an oscillating gravitational potential. We formulate the lensing response of this oscillating cloud as a discrete sideband expansion: the periodic lens redistributes each incident frequency component among discrete sidebands separated by integer multiples of the cloud’s beat frequency. Applying this framework to stellar-mass binary-black-hole inspirals lensed by supermassive black holes hosting boson clouds, we compute the dynamically lensed waveforms and compare them with their static-lens counterparts. A waveform-mismatch analysis identifies broad regions of parameter space where the dynamical modulation produces substantial, distinguishable deviations from static lensing. These signatures offer a potential probe of coherent level mixing and self-interactions in the ultralight-boson sector.