非均匀磁场作用下海森堡自旋模型的时间演化

发布时间：2018-02-26 09:12

本文关键词： 海森堡XY模型海森堡XXZ模型纠缠度时间演化　出处：《西南大学》2012年硕士论文　论文类型：学位论文

【摘要】：量子信息学是以量子力学与经典信息学想结合的一门新兴学科,它给信息科学和技术的变革、持续高速的发展提供了新的原理和方法。微观系统的量子特性为信息学带来了许多令人耳目一新的现象,其中量子纠缠就是最为神奇和最富有代表性的特征之一。研究好量子态的纠缠随时间的演化特性,能够有效地实现对固态量子系统的操控,处理量子系统中信息的存储以及信息的传输。本文主要讨论了在非均匀磁场作用下海森堡XY自旋模型和混合海森堡XXZ自旋模型的纠缠随时间演化的情况,为研究固态量子系统纠缠的动力学行为和实现其信息的操控和处理提供一些重要的理论依据和途径。首先,本文引入了Wootters关于纠缠度(Concurrence)的概念,研究了在非均匀磁场作用下的海森堡XY自旋模型的时间演化情况。通过对系统的哈密顿量进行求解,找到时间演化算符。分别选择不同的初始状态,得到了系统随时间变化的密度矩阵,从而计算得到系统的纠缠度。分析了纠缠度随时间的变化情况,讨论了各个参数对纠缠度的影响。结果发现,其纠缠度是一个随时间呈正弦变化的周期函数。耦合参数J只会影响纠缠度的周期大小,增大J可以使周期T缩短,而各向异性参数γ对纠缠程度有增强的作用。外界磁场B以及其不均匀度b都会抑制两粒子的纠缠度,因此可以通过调节外界磁场来操控系统的纠缠情况。另外,系统的初态对系统随时间演化也扮演着十分重要的角色。其次,我们研究了在非均匀磁场作用下的海森堡XXZ自旋模型的时间演化情况。首先对纠缠度的定义进行了合理推广并进行了证明,将其应用于混合自旋模型中。同样地,通过对哈密顿量的求解,找到不同初态下系统随时间演化的密度矩阵,从而得到系统的纠缠度。通过数值模拟分析纠缠度随时间的变化情况和各个参数以及系统初始状态对纠缠度的影响。研究表明,混合海森堡模型的纠缠随时间演化因不同的初态的选择而有所不同,但仍然呈正弦变化的周期函数。外界磁场B的大小对纠缠度C并没有影响,而是决定于磁场的不均匀度b,外界磁场越不均匀,对两粒子纠缠的抑制作用就越大。各向异性参数k的值也会对其纠缠状态产生影响。
[Abstract]:Quantum informatics is a new subject which combines quantum mechanics with classical informatics. It brings the revolution of information science and technology. The continuous rapid development provides new principles and methods. The quantum properties of micro systems bring many new phenomena to informatics. Quantum entanglement is one of the most magical and representative characteristics. By studying the evolution of entanglement over time, we can effectively control the quantum system in solid state. This paper mainly discusses the entanglement evolution of Heisenberg XY spin model and mixed Heisenberg XXZ spin model under the action of inhomogeneous magnetic field. This paper provides some important theoretical basis and approaches for studying the dynamical behavior of entanglement in solid-state quantum systems and realizing the manipulation and processing of its information. Firstly, the concept of entanglement degree concurrenceby Wootters is introduced, and the time evolution of Heisenberg XY spin model under the action of inhomogeneous magnetic field is studied. The Hamiltonian of the system is solved. The time evolution operator is found. The density matrix of the system varying with time is obtained by choosing different initial states, and the entanglement degree of the system is calculated. The variation of the entanglement degree with time is analyzed. The influence of each parameter on the entanglement degree is discussed. It is found that the entanglement degree is a periodic function with sine change with time. The coupling parameter J will only affect the period of entanglement degree, and increasing J can shorten the period T. The anisotropic parameter 纬 can enhance the degree of entanglement. Both the external magnetic field B and its inhomogeneity b can inhibit the entanglement degree of two particles, so the entanglement of the system can be controlled by adjusting the external magnetic field. The initial state of the system also plays a very important role in the evolution of the system over time. Secondly, we study the time evolution of Heisenberg XXZ spin model under the action of inhomogeneous magnetic field. Firstly, we generalize the definition of entanglement reasonably and prove that it is applied to the mixed spin model. By solving the Hamiltonian, the density matrix of the system evolves with time under different initial states is found. Then the entanglement degree of the system is obtained. The variation of entanglement degree with time and the influence of various parameters and the initial state of the system on the entanglement degree are analyzed by numerical simulation. The entanglement of the mixed Heisenberg model varies with time depending on the choice of the initial state, but it still presents a sinusoidal periodic function. The magnitude of the external magnetic field B has no effect on the entanglement degree C. It is determined by the inhomogeneity of the magnetic field b, the more uneven the external magnetic field, the greater the suppression of the two-particle entanglement, and the value of the anisotropic parameter k will also affect the entanglement state of the two particles.
【学位授予单位】：西南大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：O413.1;TP3

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