  Using the quadratic expansion in the photon fields of Euler-Heisenberg (EH)
non-linear electrodynamics (NLED) Lagrangian model we study relevant vacuum
properties in a scenario involving the propagation of a photon probe in the
presence of a background constant and static magnetic field, ${\bf B_e}$. We
compute the gauge invariant, symmetric and conserved energy-momentum tensor
(EMT) and angular momentum tensor (AMT) for arbitrary magnetic field strength
using the Hilbert method under the soft-photon approximation. We discuss how
the presence of magneto-electric terms in the EH Lagrangian is a source of
anisotropy, induce the non-zero trace in the EMT and leads to differences
between EMT calculated by the Hilbert or Noether method. From the Hilbert EMT
we analyze some quantities of interest such as the energy density, pressures,
Poynting vector, and angular momentum vector, comparing and discussing the
differences with respect to the improved Noether method. The magnetized vacuum
properties are also studied showing that a photon effective magnetic moment can
be defined for different polarization modes. The calculations are done in terms
of derivatives of the two scalar invariants of electrodynamics, hence,
extension to other NLED Lagrangian is straightforward. We discuss further
physical implications and experimental strategies to test magnetization, photon
pressure, and effective magnetic moment.
