  The evolution of the Von Neumann entanglement entropy of a $n$-dimensional
mirror influenced by the strongly coupled $d$-dimensional quantum critical
fields with a dynamic exponent $z$ is studied by the holographic approach. The
dual description is a $n+1$-dimensional probe brane moving in the
$d+1$-dimensional asymptotic Lifshitz geometry ended at $r=r_b$, which plays a
role as the UV energy cutoff. Using the holographic influence functional
method, we find that in the linear response region, by introducing a harmonic
trap for the mirror, which serves as a IR energy cutoff, the Von Neumann
entropy at late times will saturate by a power-law in time for generic values
of $z$ and $n$. The saturated value and the relaxation rate depend on the
parameter $\alpha\equiv 1+(n+2)/z$, which is restricted to $1<\alpha <3$ but
$\alpha \ne 2$. We find that the saturated values of the entropy are
qualitatively different for the theories with $1<\alpha<2$ and $2<\alpha<3$.
Additionally, the power law relaxation follows the rate $\propto
t^{-2\alpha-1}$. This probe brane approach provides an alternative way to study
the time evolution of the entanglement entropy in the linear response region
that shows the similar power-law relaxation behavior as in the studies of
entanglement entropies based on Ryu-Takayanagi conjecture. We also compare our
results with quantum Brownian motion in a bath of relativistic free fields.
