  Massive bosons in the vicinity of Kerr-Newman black holes can form pure bound
states when their phase angular velocity fulills the synchronisation condition,
i.e. at the threshold of superradiance. The presence of these stationary clouds
at the linear level is intimately linked to the existence of Kerr black holes
with synchronised hair at the non-linear level. These configurations are very
similar to the atomic orbitals of the electron in a hydrogen atom. They can be
labeled by four quantum numbers: $n$, the number of nodes in the radial
direction; $\ell$, the orbital angular momentum; $j$, the total angular
momentum; and $m_j$, the azimuthal total angular momentum. These synchronised
configurations are solely allowed for particular values of the black hole's
mass, angular momentum and electric charge. Such quantization results in an
existence surface in the three-dimensional parameter space of Kerr-Newman black
holes. The phenomenology of stationary scalar clouds has been widely addressed
over the last years. However, there is a gap in the literature concerning their
vector cousins. Following the separability of the Proca equation in
Kerr(-Newman) spacetime, this work explores and compares scalar and vector
stationary clouds around Kerr and Kerr-Newman black holes, extending previous
research.
