Plasma modes in QED super-strong magnetic fields of magnetars and laser plasmas

Ultra-magnetized plasmas, where the magnetic field strength exceeds the Schwinger field of about $B_{Q}\approx4\times10^{13}$~gauss, become of great scientific interest, thanks to the current advances in laser-plasma experiments and astrophysical observations of magnetar emission. These advances demand better understanding of how quantum electrodynamics (QED) effects influence collective plasma phenomena. In particular, Maxwell's equations become nonlinear in the strong-QED regime. Here we present the `QED plasma framework' which will allow one to {\em systematically} explore collective phenomena in a QED-plasma with arbitrarily strong magnetic field. Further, we illustrate the framework by exploring low-frequency modes in the ultra-magnetized, cold, electron-positron plasmas. We demonstrate that the classical picture of five branches holds in the QED regime; no new eigenmodes appear. The dispersion curves of all the modes are modified. The QED effects include the overall modification to the plasma frequency, which becomes field-dependent. They also modify resonances and cutoffs of the modes, which become both field- and angle-dependent. The strongest effects are (i) the {\em field-induced transparency of plasma} for the O-mode via the dramatic reduction of the low-frequency cutoff well below the plasma frequency, (ii) the {\em Alfven mode suppression} in the large-$k$ regime via the reduction of the Alfven mode resonance, and (iii) the {\em O-mode slowdown} via strong angle-dependent increase of the index of refraction. These results should be important for understanding of a magnetospheric pair plasma of a magnetar and for laboratory laser-plasma experiments in the QED regime.