pattern of the observed spectrum will be affected by the presence of ALPs.

The purpose of this paper is to investigate the changes in polarization patterns in the observed X-ray spectra of magnetars in the presence of ALPs. Our approach consists of the following steps. Firstly, we assume that for every frequency $\omega\sim1-\mathcal{O}(\text{few hundred})\,\text{keV}$, both photons and ALPs are produced near the surface of the magnetar, in an uncorrelated manner. The important observation is that the astrophysical processes leading to photon production both in the soft as well as the hard X-ray regime are completely independent of the processes that give rise to ALPs, mainly by nucleon-nucleon bremsstrahlung [@Iwamoto:1984ir; @Iwamoto:1992jp; @Nakagawa:1987pga; @Nakagawa:1988rhp; @Raffelt:1996wa; @Umeda:1997da; @Paul:2018msp; @Maruyama:2017xzl].

The evolution of the ALP-photon system in the magnetic field of the magnetar is conveniently described in terms of a set of first-order differential equations coupling the amplitudes and phase difference of the ALP and photon fields. This parametrization has been shown to drastically simplify numerical analyses of the system [@Fortin:2018ehg]. The uncorrelated production mechanism of ALPs and photons alluded to above allows us to average over the initial phase differences. On the other hand, we parametrize the initial relative amplitudes by the angle 