Crab Pulsar: IXPE Observations Reveal Unified Polarization Properties Across Optical and Soft X-Ray Bands

We present a phase-dependent analysis of the polarized emission from the Crab pulsar based on three sets of observations by the Imaging X-ray Polarimetry Explorer (IXPE). We found that a phenomenological model involving a simple linear transformation of the Stokes parameters adequately describes the IXPE data. This model enables us to establish, for the first time, a connection between the polarization properties of the Crab pulsar in the optical and soft X-ray bands, suggesting a common underlying emission mechanism across these bands, likely synchrotron radiation. In particular, the phase-dependent polarization degree in X-rays for pure pulsar emission shows similar features but is reduced by a factor $\approx (0.46-0.56)$ compared to the optical band (accounting for the contribution of the Knot in the optical), implying an energy-dependent polarized emission. In addition, using this model, we study the polarization angle swing in X-rays and identify a potentially variable phase-shift at the interpulse relative to the optical band, alongside a phase-shift marginally consistent with zero persisting at the main pulse. While the origin of this variability is unknown and presents a new challenge for theoretical interpretation, our findings suggest that the emission mechanism for the main pulse is likely located far from the neutron star surface, perhaps near or beyond the light cylinder, rather than operating in the inner magnetosphere where vacuum birefringence is expected to be at work. Ignoring the phase-shifts would result in identical phase-dependent polarization angles between the optical and X-ray bands for pure pulsar emission.