$\chi_0$. For example, $\chi_0=0$ signifies a pure ALP initial state. The introduction of $\chi_0$ allows us to remain agnostic about astrophysical models of production of X-ray photons near a magnetar’s surface.

Our second step is to define an astrophysics-subtracted (or surface-subtracted) normalized Stokes parameter $R$. This is done as follows. The sum ($I=\bar{I}_\perp+\bar{I}_\parallel$) and difference ($Q=\bar{I}_\perp-\bar{I}_\parallel$) of the phase-averaged photon intensities in the parallel and perpendicular planes are first computed, and then the quantities $\Delta I$ and $\Delta Q$ are defined, which are the values of the respective Stokes parameters away from the magnetar, minus their values at the magnetar’s surface $r_0$. This is very useful, since at the surface (which we take to also include the plasma) astrophysical effects due to differential opacities of the plasma lead to polarization, which should be subtracted away to extract the contribution to the polarization coming purely from ALP-photon mixing in the magnetosphere. In the absence of ALPs (*i.e.* if ALPs do not exist in the Universe), and assuming that astrophysical processes in the magnetosphere do not lead to substantial polarization, the surface-subtracted Stokes parameters should vanish.

We find, remarkably, that the surface-subtracted Stokes parameters $\Delta I$ and $\Delta Q$ factorize into two 