  Exploring and understanding ultrafast processes at the atomic level is a
scientific challenge. Femtosecond X-ray Absorption Spectroscopy (XAS) is an
essential experimental probing technic, as it can simultaneously reveal both
electronic and atomic structures, and thus unravel their non-equilibrium
dynamic interplay which is at the origin of most of the ultrafast mechanisms.
However, despite considerable efforts, there is still no femtosecond X-ray
source suitable for routine experiments. Here we show that betatron radiation
from relativistic laser-plasma interaction combines ideal features for
femtosecond XAS. It has been used to investigate the non-equilibrium transition
of a copper sample brought at extreme conditions of temperature and pressure by
a femtosecond laser pulse. We measured a rise time of the electron temperature
below 100 fs. This first experiment demonstrates the great potential of the
betatron source and paves the way to a new class of ultrafast experiments.
