  We study the imprints of new ultralight particles on the gravitational-wave
signals emitted by binary black holes. Superradiant instabilities may create
large clouds of scalar or vector fields around rotating black holes. The
presence of a binary companion then induces transitions between different
states of the cloud, which become resonantly enhanced when the orbital
frequency matches the energy gap between the states. We find that the time
dependence of the orbit significantly impacts the cloud's dynamics during a
transition. Following an analogy with particle colliders, we introduce an
S-matrix formalism to describe the evolution through multiple resonances. We
show that the state of the cloud, as it approaches the merger, carries vital
information about its spectrum via time-dependent finite-size effects.
Moreover, due to the transfer of energy and angular momentum between the cloud
and the orbit, a dephasing of the gravitational-wave signal can occur which is
correlated with the positions of the resonances. Notably, for intermediate and
extreme mass ratio inspirals, long-lived floating orbits are possible, as well
as kicks that yield large eccentricities. Observing these effects, through the
precise reconstruction of waveforms, has the potential to unravel the internal
structure of the boson clouds, ultimately probing the masses and spins of new
particles.
