  Gravitational signatures of black hole superradiance are a unique probe of
ultralight particles that are weakly-coupled to ordinary matter. The existence
of an ultralight boson would lead spinning black holes with size comparable to
the Compton wavelength of the boson to become superradiantly unstable to
forming an oscillating cloud, spinning down the black hole, and radiating
gravitational waves in the process. However, maximizing the chance of observing
such signals or, in their absence, placing the strongest constraints on the
existence of such particles, requires accurate theoretical predictions. In this
work, we introduce a new gravitational waveform model, SuperRad, that models
the dynamics, oscillation frequency, and gravitational wave signals of these
clouds by combining numerical results in the relativistic regime with fits
calibrated to analytical estimates, covering the entire parameter space of
ultralight scalar and vector clouds with the lowest two azimuthal numbers ($m =
1$ and $2$). We present new calculations of the gravitational wave frequency
evolution as the boson cloud dissipates, including using fully
general-relativistic methods to quantify the error in more approximate
treatments. Finally, as a first application, we assess the viability of
conducting follow-up gravitational wave searches for ultralight vector clouds
around massive black hole binary merger remnants. We show that LISA may be able
to probe vector masses in the range from $1\times 10^{-16}$ eV to $6\times
10^{-16}$ eV using follow-up gravitational wave searches.
