  Superradiant instability of rotating black holes (BHs) leads to the formation
of a cloud of ultralight bosons, such as axions. When the BH with the cloud
belongs to a binary system and is in an inspiraling orbit, the resonant
transition between the axion's bound states can occur. We study the history of
the evolution of the binary system accompanying the cloud composed of the
fastest growing mode, and its impact on the observational signatures,
especially for small mass ratio cases. In this case, the hyperfine resonance,
which has a very small resonance frequency, is relevant. Therefore, due to the
long timescale, we should take into account the decaying process of axions in
the transition destination mode, the backreaction to the orbital motion and the
central BH, and gravitational emission from the cloud. We present a formulation
to examine the evolution of the system around the resonance and useful
expressions for the analysis. As a result, we found the mass of the cloud that
can remain after the resonance is, at most, about $10^{-5}$ of the central BH.
The maximum remaining cloud mass is achieved when the mass ratio of the binary
is $q\sim10^{-3}$. In addition, we show that the resonant transition hardly
changes the BH mass and spin distribution, while the associated modification of
the gravitational wave frequency evolution when the binary pass through the
resonance can be a signature of the presence of the cloud.
