  We consider an axion cloud around a black hole with background magnetic
fields. We calculate the decay rate of the axion cloud due to the axion-photon
conversion associated with the axion-photon coupling. For simplicity, we
consider the situation where the axion configuration is dominated by a solution
for the eigenvalue equation equivalent to that for the Hydrogen atom, and the
coupling term can be evaluated by a successive perturbation method. For the
monopole background, we find the decay rate of the axion cloud is given by
$\sim q^2\kappa^2(GM)^5\mu^8$, where $\mu$, $M$, $G$, $\kappa$ and $q$ are the
axion mass, black hole mass, gravitational constant, coupling constant of the
axion-photon coupling and monopole charge, respectively. For the uniform
background magnetic field, we obtain the decay rate of the axion cloud $\sim
B_0^2\kappa^2 (GM)^7\mu^6$, where $B_0$ is the magnetic field strength.
Applying our formula to the central black hole in our galaxy, we find that the
value of the decay rate for the case of the uniform magnetic field is
comparable to the growth rate of the superradiant instability with $\kappa\sim
10^{-12}{\rm GeV^{-1}}$, $B_0\sim 10^3{\rm G}$ and $\mu\sim 10^{-18}{\rm eV}$.
The ratio is $10^5$ times larger for the monopole magnetic field with the same
values of the parameters.
