$xr_0$ from the magnetar, the relevant Stokes parameters are $I$ and $Q$ which correspond respectively to the sum of and the difference between the averaged perpendicular photon intensity and the averaged parallel photon intensity, *i.e.* \[EqnStokes\] As pointed out before, since the perpendicular photon field does not mix with the ALP-parallel photon two-state system, its averaged intensity is constant. Therefore, all modifications to the Stokes parameter are driven by the averaged parallel photon intensity.

Moreover, since plasma contributions are negligible outside the magnetar’s atmosphere, in the absence of ALPs the evolution equations correspond to two independent one-state systems, one for the parallel photon field and one for the perpendicular photon field. Hence without ALPs the Stokes parameters at a distance $xr_0$ from the magnetar would be the same as the Stokes parameters at the magnetar’s surface. The effects of the existence of ALPs and the possible photon oscillations with them on the Stokes parameters can therefore be conveniently analysed by studying the differences between the Stokes parameters at $xr_0$ and the Stokes parameters at $r_0$, *i.e.* \[EqnStokesDiff\] The relation $\Delta Q(\chi_0,x)=-\Delta I(\chi_0,x)$ demonstrates perfect anti-correlation between the surface-subtracted Stokes parameter $\Delta I(\chi_0,x)$ and the surface-subtracted Stokes parameter $\Delta Q(\chi_0,x)$. Therefore, 