Non-linear interaction of laser light with vacuum: contributions to the energy density and pressure in the presence of an intense magnetic field

Recent simulations show that very large electric and magnetic fields near the kilo tesla strength will likely be generated by ultra-intense lasers at existing facilities over distances of hundreds of microns in underdense plasmas. Even stronger ones are expected in the future although some technical difficulties must be overcome. In addition, it has been shown that vacuum exhibits a peculiar non-linear behaviour in the presence of high magnetic and electric field strengths. In this work, we are interested in the analysis of the thermodynamical contributions of vacuum to the energy density and pressure when radiation interacts with it in the presence of an external magnetic field. Using the Euler-Heisenberg formalism in the regime of weak fields i.e. smaller than critical quantum electrodynamics field strength values, we evaluate these magnitudes and analyse the highly anisotropic behaviour we find. Our work has implications for photon-photon scattering with lasers and astrophysically magnetized underdense systems far outside their surface where matter effects are increasingly negligible.