been no such factorization. Hence, if there exists a common origin for the production of ALPs and photons, then the initial phase difference would be fixed and the dependence on $\chi_0$ would not be as straightforward as in .

Finally, since the argument leading to the results above relies only on the probability conservation property of the system, it is important to note that relations analogous to for averaged intensities exist for any two-state oscillation system satisfying the probability conservation property. Although we have not investigated it further, we expect that relations analogous to for averaged intensities can be found for any $n$-state oscillation system as long as probability is conserved.

Results {#SResults}
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In this section we solve numerically the evolution equations with dipole magnetic field (valid a few $r_0$ away from the magnetar) for magnetar parameters corresponding to SGR 1806-20 in the soft and hard X-ray range. As mentioned in the introduction, we stay agnostic with respect to the different production mechanisms for ALPs and photons. Our results are therefore conveniently expressed with the help of the normalized surface-subtracted Stokes parameter far away from the magnetar where the magnetic field is effectively turned off.

In principle, forces us 