  For short wavelengths, it is well known that the linearized Wigner-Moyal
equation predicts wave damping due to wave-particle interaction, where the
resonant velocity shifted from the phase velocity by a velocity $v_q = \hbar
k/2m$. Here $\hbar$ is the reduced Planck constant, $k$ is the wavenumber and
$m$ is the electron mass. Going beyond linear theory, we find additional
resonances with velocity shifts $n v_q$, $n = 2, 3, \ldots$, giving rise to a
new wave-damping mechanism that we term \emph{multi-plasmon damping}, as it can
be seen as the simultaneous absorption (or emission) of multiple plasmon
quanta. Naturally this wave damping is not present in classical plasmas. For a
temperature well below the Fermi temperature, if the linear ($n = 1$) resonant
velocity is outside the Fermi sphere, the number of linearly resonant particles
is exponentially small, while the multi-plasmon resonances can be located in
the bulk of the distribution. We derive sets of evolution equations for the
case of two-plasmon and three-plasmon resonances for Langmuir waves in the
simplest case of a fully degenerate plasma. By solving these equations
numerically for a range of wave-numbers we find the corresponding damping
rates, and we compare them to results from linear theory to estimate the
applicability. Finally, we discuss the effects due to a finite temperature.
