Probing vacuum birefringence under a high-intensity laser field with gamma-ray polarimetry at the GeV scale

Probing vacuum structures deformed by high intense fields is of great interest in general. In the context of quantum electrodynamics (QED), the vacuum exposed by a linearly polarized high-intensity laser field is expected to show birefringence. We consider the combination of a 10 PW laser system to pump the vacuum and 1 GeV photons to probe the birefringent effect. The vacuum birefringence can be measured via the polarization flip of the probe $\gamma$-rays which can also be interpreted as phase retardation of probe photons. We provide theoretically how to extract phase retardation of GeV probe photons via pair-wise topology of the Bethe-Heitler process in a polarimeter and then evaluate the measurability of the vacuum birefringence via phase retardation given a concrete polarimeter design with a realistic set of laser parameters and achievable pulse statistics.