The Time Evolution Of Entanglement In Heisenberg Spin Chain

Quantum entanglement is an important resource of quantum information science. Using the time evolution properties of entanglement in quantum systems, the problems, such as the information storage and information transmission, can be effectively processed. The quantum control can be achieved. In this paper, the time evolution of the entanglement of a pair of two spin particles under the influence of the Heisenberg spin chain is analyzed. The free time evolution of the entanglement in a Heisenberg spin chain is also discussed.First, a model of a pair of coupled particles interacting with a Heisenberg spin chain is presented. A pair of spin particles is coupled symmetrically on both sides of a ring composed of spin particles. By symmetry analysis and perturbation theory, the effective Hamiltonian of the system is derived. The time evolution of entanglement of the pair particles is calculated. For a ring of two spins, the entanglement between the two spin particles can be created and is a periodic function of time when the external magnetic field is greater than a critical value. If the number of spins in the ring is large, the period of the entanglement changes anomalously with the external magnetic field. When a Bell state is chosen as the ground state, the entanglement can be stored with relative large value in a range of external magnetic field. There appear regular jumps in the entanglement when the number of spins in the ring is large. The number of jumps is the half of the spins in the ring.Next, the free time evolution of the entanglement in a Heisenberg XY spin chain is discussed. Through analytic method and numerical simulation, the time evolution of the entanglement is obtained. The relation between the entanglement and the anisotropic parameter, the magnetic field is analyzed. The entanglement is an approximately linear increase with increasing the anisotropy. If the magnetic field is fixed, the time evolution of the entanglement is periodic and is a sinusoidal function. When number of spins increases, the entanglement is also periodic but the next-nearest neighbor will affect the entanglement. When the magnetic field is varied with small magnetic field, the effects of the next nearest neighbor exceeds that of the magnetic field, the entanglement appears irregular behavior. When the magnetic field is very large, the effects of the magnetic field exceeds that of the next nearest neighbor, the entanglement becomes again regular and periodic.