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

Abstract

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 < α < 3 but α = 2. We find that the saturated values of the entropy are qualitatively different for the theories with 1 < α < 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.Entanglement entropies provide useful probes to non-local properties of quantum systems, which are important for understanding quantum phase transitions [1], and moreover are the key quantities in quantum information processing [2]. The idea of entanglement entropies has also received much attention in connection to the information paradox in black hole physics [3]. It is generally impossible to isolate a particular quantum system in which we are interested from its surrounding. Considering a full theory that describes the interaction between system and environment, from which all information like correlation functions can in principle be obtained. Tracing or integrating out the degrees of freedom of the environment to obtain an effective field theory for the system leads to a loss of information. If the quantum state in the full theory is a pure state, namely a zero entropy state, tracing out the environmental degrees of freedom yields a reduced density matrix for the degrees of freedom of the system, which typically becomes a mixed state with non-vanishing entropy. The Von-Neumann entropy is a measure of the loss of information in the process of integrating out some degrees of freedom in the full unitary system [4,5]. The effective theory can be described by the reduced density matrix ρ r , which is obtained by tracing out the environmental variables in the full density matrix using the method of Feyman-Vernon influence functional [6]. The Von Neumann entropy is then defined by −T rρ r ln ρ r with the trace over system's variables. In general, the effective theory for the system obtained in this way is not unitary, and it can lead to the dissipative and stochastic...

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