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 α ≡ 1+(n+2)/z, which is restricted to 1 < 3 but α = 2. We find that the saturated values of the entropy are qualitatively different for the theories with 1 < 2 and 2 < α < 3. Additionally, the power law relaxation follows the rate ∝ t^−2α−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.