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Gravitational waves (GWs) are an exciting new probe of physics beyond the standard models of gravity and particle physics. One interesting possibility is provided by the so-called "gravitational atom," wherein a superradiant instability spontaneously forms a cloud of ultralight bosons around a rotating black hole. The presence of these boson clouds affects the dynamics of black hole binary inspirals and their associated GW signals. In this Letter, we show that the binary companion can induce transitions between bound and unbound states of the cloud, effectively "ionizing" it, analogous to the photoelectric effect in atomic physics. The orbital energy lost in this process can overwhelm the losses due to GW emission, so that ionization drives the inspiral rather than merely perturbing it. We show that the ionization power contains sharp features that lead to distinctive "kinks" in the evolution of the emitted GW frequency. These discontinuities are a unique signature of the boson cloud and observing them would not only constitute a detection of the ultralight boson itself, but also provide direct information about its mass and the state of the cloud.