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We investigate the time-dependent Rényi entanglement entropy after a quantum quench starting from the Mott-insulating and charge-density-wave states in a one-dimensional free boson system. The second Rényi entanglement entropy is found to be the negative of the logarithm of the permanent of a matrix consisting of time-dependent single-particle correlation functions. From this relation and a permanent inequality, we obtain rigorous conditions for satisfying the volume-law entanglement growth. We also succeed in calculating the time evolution of the entanglement entropy in unprecedentedly large systems by brute-force computations of the permanent. We discuss possible applications of our findings to the real-time dynamics of noninteracting bosonic systems.

We study dynamics of entanglement in the thermalization process of an isolated quantum manybody system. We propose a simple setup for measuring the propagation speed of entanglement entropy (EE) in numerical simulations and apply it to the integrable/non-integrable spin models in 1D -the transverse Ising (TI) model, the chaotic Ising (CI) model, and the extended chaotic Ising (ECI) model. We find that two distinct time-scales t * and t diff arise in the dynamics of EE in the thermalization process: the former represents the time-scale for the saturation of EE and the latter characterizes spreading of EE over the entire system. Evaluating the propagation speed of entanglement from t diff , we find that entanglement propagates ballistically with a constant velocity irrespective of the integrability of the model. The propagation speed of entanglement is found to coincide with the maximum group velocity of quasi-particle excitations in the TI model. We also evaluate the propagation speed of entanglement by mutual information and find that it agrees well with the one evaluated by EE. We discuss the condition for thermalization based on the numerical results and propose that scrambling of the entire system has to take place before saturation of EE for thermalization.

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We study the time evolution of entanglement entropy of bosons in a one-dimensional optical lattice induced by a sudden quench of the hopping amplitude J. We consider the system being quenched into the deep Mott-insulating (MI) regime, i.e., J/U 1 (U is the strength of the on-site repulsive interaction), from the product state with individual boson isolated in each lattice site. The lowenergy excited states in this regime can be effectively described by fermionic quasiparticles known as doublons and holons. Developing the effective theory, we analytically calculate the time evolution of the second-order Rényi entropy (RE) for a subsystem and propose a quasiparticle picture for the time evolution of the RE based on the obtained analytic expressions. Doublons and holons are excited by the quench as entangled pairs that propagate with the velocity vpair = 6J. The RE reflects the population of doublon-holon pairs that span the boundary of the subsystem. In the short-time scale [Jt/ = O(1)], the RE exhibits the rapid oscillations with the frequency U/ , while in the long-time scale (Jt/ 1) the RE grows linearly in time until the pair spreads beyond the size of the subsystem and the RE saturates to a constant. We find that doublon-holon pairs are emitted periodically in time after the quench and those emitted at t > 0 explain the smooth transition between the linear growth and the saturation of the RE.