Accretion Disk Perturbations and Their Effects on Kerr Black Hole Superradiance and Gravitational Atom Evolution

Kerr black hole (BH) superradiance can form gravitational atoms and produce characteristic gravitational-wave signals, providing a probe of ultralight bosons and dark matter. In realistic systems, accretion-disk gravity can shift energy levels and mix states, modifying the effective superradiant growth. We model the disk as a weak external perturbation via a multipole expansion and derive an effective three-level Hamiltonian for the $n=2$ subspace $\{\ket{211},\ket{210},\ket{21-1}\}$ in the weak-coupling regime. The leading disk effect is the quadrupolar ($\ell_d=2$) tidal field, whose symmetries fix the selection rules: axisymmetry gives only diagonal shifts, equatorial nonaxisymmetry activates $\Delta m=\pm2$ mixing ($\ket{211}\leftrightarrow\ket{21-1}$), and breaking equatorial reflection opens $\Delta m=\pm1$ couplings involving $\ket{210}$. As illustrations, a transient equatorial $m=2$ spiral wave drives the resulting two-level system and can suppress superradiance by populating a decaying mode, while a quasi-static warp produces full three-level mixing and can generate narrow ``growth gaps'' near accidental near-degeneracies, with the same static reshuffling also allowing enhancement when weight shifts toward the growing mode. These findings demonstrate that accretion disk perturbations are a crucial environmental factor in determining the dynamics of BH superradiance and the evolution of boson clouds, thereby providing a more reliable theoretical basis for assessing the detectability of ultralight bosons in realistic astrophysical settings.