Time evolution of entanglement entropy of moving mirrors influenced by strongly coupled quantum critical fields

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.