  A new generation of coronagraphs to study the solar wind and CMEs are being
developed and launched. These coronagraphs will heavily rely on multi-channel
observations where visible light (VL) and UV-EUV observations provide new
plasma diagnostics. One of these instruments, Metis on board ESA-Solar Orbiter,
will simultaneously observe VL and the UV Lyman-$\alpha$ line. The number of
neutral Hydrogen atoms (a small fraction of coronal protons) is a key parameter
for deriving plasma properties such as temperature from the observed
Lyman-$\alpha$ line intensity. However, these measurements are significantly
affected if non-equilibrium ionisation effects occur. The aim of this work is
to determine if non-equilibrium ionisation effects are relevant in CMEs and in
particular when and in which regions of the CME. We use a magneto-hydrodynamic
simulation of a magnetic flux rope ejection to generate a CME. From this we
then reconstruct the ionisation state of Hydrogen atoms in the CME by
evaluating both the advection of neutral and ionised Hydrogen atoms and the
ionisation and recombination rates in the MHD simulation. We find that the
equilibrium ionisation assumption holds mostly in the core of the CME. In
contrast non-equilibrium ionisation effects are significant at the CME front,
where we find about 100 times more neutral Hydrogen atoms than prescribed by
ionisation equilibrium conditions, even if this neutral Hydrogen excess might
be difficult to identify due to projection effects. This work provides key
information for the development of a new generation of diagnostic techniques
that aim at combining visible light and Lyman-$\alpha$ line emissions. The
results show that non-ionisation equilibrium effects need to be considered when
we analyse CME fronts. To incorrectly assume equilibrium ionisation in these
regions would lead to a systematic underestimate of plasma temperatures.
