to compute $R$ by averaging over the initial phase difference for several mixed initial states. However, the relation allows us to focus only on the ALP-to-photon conversion probability for pure ALP initial state. Nevertheless, to verify the relation , we compute (for various ALP and magnetar parameters) $R$ the hard way using and compare with the ALP-to-photon conversion probability for pure ALP initial state through , validating the argument of the previous section.

For SGR 1806-20, the ALP-to-photon conversion probability for pure ALP initial state in the soft and hard X-ray range was studied in [@Fortin:2018ehg]. For completeness, we include here the relevant plots as well as a discussion of the relevant features of $R(\chi_0=0)\equiv R(\chi_0=0,x=\infty)$ for a benchmark point that is not excluded by CAST [@Anastassopoulos:2017ftl].

Dependence on the ALP and Magnetar Parameters
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As mentioned previously, standard astrophysical considerations dictates how the magnetar and its atmosphere behave. Hence, theoretical astrophysics determines the initial amplitudes of the particles at the magnetar’s surface. The particles then travel through the plasma in the magnetar’s atmosphere, which is of the order of centimeters, where they can encounter a resonance in the parallel-perpendicular photon space [@Lai:2006af]. Since we remain agnostic about the astrophysical 