$Q$ in Fig. \[FigQ\]. The astrophysical prediction of $Q$ is expected to be $Q\sim\bar{I}_\perp$, while the presence of ALP-to-photon conversion drives $Q$ to be smaller, and perhaps even negative depending on the intensity of ALPs produced from the core. The next generation of X-ray polarimeters [@polstar; @ixpe] and increasingly sophisticated modeling of the astrophysics of magnetars [@Lai:2006af; @Chelouche:2008ta; @Jimenez:2011pg; @Perna:2012wn; @Wadiasingh:2017rcq] provide an opportunity to investigate ALPs using polarization.

Our paper is organized as follows. In Section \[SOsc\], we recapitulate the ALP-photon coupled system and the results of [@Fortin:2018ehg]. In Section \[SPP\], we perform the calculation of the Stokes parameters. In Section \[SResults\], we find analytical approximations for the parameter $R$ and display it as a function of the photon energy and ALP-photon coupling. We also describe the observational possibilities and contrast our results with previous work in Section \[SObs\]. We end with our conclusions in Section \[SConclusion\]. Finally, Appendix \[SApp\] demonstrates the factorization property of the normalized surface-subtracted Stokes parameter $R$.

Oscillations {#SOsc}
============

In this section we introduce the evolution equations for the photon and the ALP relevant to magnetars. We use the general formalism of [@Fortin:2018ehg] for oscillations in the limits where the space variations of the 