two independent oscillation systems, a two-state system for the ALP and the parallel photon, and a one-state system for the perpendicular photon.

Due to the probability conservation property $\frac{d}{dx}[|a(x)|^2+|E_\parallel(x)|^2]=0$ discussed in [@Fortin:2018ehg], the different states can be expressed as \[EqnaEE\] where $A_a=A\cos[\chi(x)]$, $A_\parallel=A\sin[\chi(x)]$ and $A_\perp$ are the amplitudes at position $xr_0$ of the ALP field, the parallel photon field and the perpendicular photon field respectively. It is important to note that $A$ and $A_\perp$ are constants which can always be chosen real and positive while $\chi(x)$, $\phi_a(x)$, $\phi_\parallel(x)$ and $\phi_\perp(x)$ are real functions. Hence, the intensities at position $xr_0$ are given by the respective amplitudes squared, *i.e.* $I_a(x)=A^2\cos^2[\chi(x)]$, $I_\parallel(x)=A^2\sin^2[\chi(x)]$ and $I_\perp(x)=A_\perp^2$ for the ALP field, the parallel photon field and the perpendicular photon field. Probability conservation then implies that $I_a(x)+I_\parallel(x)=A^2$ and $I_\perp(x)=A_\perp^2$ are constants.

Using in , the evolution equations become \[EqnEE\] where $\Delta\phi(x)=\phi_a(x)-\phi_\parallel(x)$ is the phase difference between the ALP field and the parallel photon field while $\Sigma\phi(x)=\phi_a(x)+\phi_\parallel(x)$ is the sum of the phases of the ALP field and the parallel photon field. In addition to showing that the differential equation for $\phi_\perp(x)$ decouples, the evolution equations imply also that the differential equation for $\Sigma\phi(x)$ decouples in the sense 