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Time Evolution of Entanglement for Heisenberg Model under Nonuniform External Magnetic Field

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Tutor: RanYangQiang
School: Southwestern University
Course: Theoretical Physics
Keywords: Heisenberg XY model,Heisenberg XXZ model,Concurrence,Time evolution
CLC: O413.1
Type: Master's thesis
Year:  2012
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Quantum informatics is a new subject of the combination which based on quantum mechanics and classical informatics. It provides new principle and method for the revolution of information science and technology and consistently high development. The quantum properties of microscopic system brought many refreshing and new phenomenon for informatics. Among them, quantum entanglement is one of the characteristics of representatives which has the most amazing and the richest can effectively achieve the controling of solid state quantum system and deal with the information storage and transmission of the quantum system.This paper mainly discusses the situation about the Heisenberg XY and anisotropic Heisenberg XXZ spin model with time evolution of entanglement in the non-uniform magnetic field. This research provides some important theoretical basis and the way for entanglement of dynamic characteristics of the solid-state quantum system and the realization of its information handling and processing.First, this paper introduced the concept of Concurrence. Then,The research studied the time evolution of Heisenberg XY spin model in the uniform magnetic field. Through the system Hamiltonian¡¯s solution can find the time evolution operator. And the different initial states were selected, the density matrix of this system was obtained, then the concurrence can be calculated easily. Selectiong different initial states can gain the density matrix of this system, and then we can get the system of entanglement through the Calculation. This essay also analyzed the situation of the changing with time and discussed the influnece about the various parameters on the degree of entanglement. The results found that the entanglement is a sine function over time. The interaction parameter J only affects the degree of entanglement cycle size, increaseing J can make cycles T shorter. But the anisotropic parameters¦Ãstrengthens the degree of entanglement. External magnetic fields B and its nonuniformity b will restrain entanglement of two particle, So we can control external magnetic field to adjust to the system¡¯s entanglement. In addition, the original state of the system also plays a very important role in the system evolution over time.Then, we studied the time evolution of entanglement for the anisotropic Heisenberg XXZ spin model under nonuniform external magnetic field. Firstly,having the reasonably extended and proved about the definition of concurrence, and The time evolution operator can be obtained by calculating the Hamilton of this model. Similarly,through the selection of the different initial states find the different initial states which under the system along with the time evolution of the density matrix, then got the system degree of entanglement. Through the numerical simulation analysis of entanglement changes over time and the various parameters as well as system initial states on the entanglement degree of the effect. The research shows that the concurrence of the anisotropic Heisenberg model differs somewhat because of different initial state¡¯s choice, but still a sine change periodic function evolve over time. The size of the external magnetic field doesn¡¯t influence the entanglement, but its nonuniformity b is decided to the degree. The more uniform the external magnetic field is, the greater inhibition is for the entanglement of the two particles. We also find that the anisotropic parameters k influences its own entangled state.
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