本文關(guān)鍵詞： 海森堡XY模型 海森堡XXZ模型 糾纏度 時間演化　出處：《西南大學(xué)》2012年碩士論文　論文類型：學(xué)位論文

【摘要】：量子信息學(xué)是以量子力學(xué)與經(jīng)典信息學(xué)想結(jié)合的一門新興學(xué)科,它給信息科學(xué)和技術(shù)的變革、持續(xù)高速的發(fā)展提供了新的原理和方法。微觀系統(tǒng)的量子特性為信息學(xué)帶來了許多令人耳目一新的現(xiàn)象,其中量子糾纏就是最為神奇和最富有代表性的特征之一。研究好量子態(tài)的糾纏隨時間的演化特性,能夠有效地實現(xiàn)對固態(tài)量子系統(tǒng)的操控,處理量子系統(tǒng)中信息的存儲以及信息的傳輸。本文主要討論了在非均勻磁場作用下海森堡XY自旋模型和混合海森堡XXZ自旋模型的糾纏隨時間演化的情況,為研究固態(tài)量子系統(tǒng)糾纏的動力學(xué)行為和實現(xiàn)其信息的操控和處理提供一些重要的理論依據(jù)和途徑。 首先,本文引入了Wootters關(guān)于糾纏度(Concurrence)的概念,研究了在非均勻磁場作用下的海森堡XY自旋模型的時間演化情況。通過對系統(tǒng)的哈密頓量進行求解,找到時間演化算符。分別選擇不同的初始狀態(tài),得到了系統(tǒng)隨時間變化的密度矩陣,從而計算得到系統(tǒng)的糾纏度。分析了糾纏度隨時間的變化情況,討論了各個參數(shù)對糾纏度的影響。結(jié)果發(fā)現(xiàn),其糾纏度是一個隨時間呈正弦變化的周期函數(shù)。耦合參數(shù)J只會影響糾纏度的周期大小,增大J可以使周期T縮短,而各向異性參數(shù)γ對糾纏程度有增強的作用。外界磁場B以及其不均勻度b都會抑制兩粒子的糾纏度,因此可以通過調(diào)節(jié)外界磁場來操控系統(tǒng)的糾纏情況。另外,系統(tǒng)的初態(tài)對系統(tǒng)隨時間演化也扮演著十分重要的角色。 其次,我們研究了在非均勻磁場作用下的海森堡XXZ自旋模型的時間演化情況。首先對糾纏度的定義進行了合理推廣并進行了證明,將其應(yīng)用于混合自旋模型中。同樣地,通過對哈密頓量的求解,找到不同初態(tài)下系統(tǒng)隨時間演化的密度矩陣,從而得到系統(tǒng)的糾纏度。通過數(shù)值模擬分析糾纏度隨時間的變化情況和各個參數(shù)以及系統(tǒng)初始狀態(tài)對糾纏度的影響。研究表明,混合海森堡模型的糾纏隨時間演化因不同的初態(tài)的選擇而有所不同,但仍然呈正弦變化的周期函數(shù)。外界磁場B的大小對糾纏度C并沒有影響,而是決定于磁場的不均勻度b,外界磁場越不均勻,對兩粒子糾纏的抑制作用就越大。各向異性參數(shù)k的值也會對其糾纏狀態(tài)產(chǎn)生影響。
[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 緯 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.
【學(xué)位授予單位】：西南大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2012
【分類號】：O413.1;TP3

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