magnetic field are large compared to the particle wavelength and dispersion is weak. The reader interested in the details of this formalism is referred to [@Fortin:2018ehg].

Evolution Equations
-------------------

For particles with energies in the $1$ to $200$ keV range (corresponding to soft and hard X-ray photons) propagating radially outwards from a magnetar, the system is in the appropriate limits (as long as the magnetic field is not too large) with negligible plasma contributions to the evolution equations [@Raffelt:1987im; @Lai:2006af], leading to \[EqnDiffMat\] where Here $a(x)$, $E_\parallel(x)$ and $E_\perp(x)$ are the ALP and parallel and perpendicular photon electric fields respectively while $x=r/r_0$ with $r$ the distance from the center of the magnetar and $r_0$ the magnetar’s radius. Moreover, $\omega$ is the energy of the particles, $m_a$ is the ALP mass, $g$ is the ALP-photon coupling constant and $\theta$ is the angle between the magnetic field and the direction of propagation of the particle. $q_\parallel$ and $q_\perp$ are dimensionless functions of the magnetic field $B$ given by [@Lai:2006af; @Raffelt:1987im] with $b=B/B_c$ where $B_c=m_e^2/e=4.414\times10^{13}\,\text{G}$ is the critical QED field strength. Here $e=\sqrt{4\pi\alpha}$ where the fine structure constant is approximatively $\alpha\approx1/137$.

Since the plasma contributions are negligible, the three-state system effectively decomposes into 