量子Lyapunov控制的理论研究及应用

发布时间：2018-11-16 15:59

【摘要】：作为一门新兴的交叉学科,量子控制论是实现量子计算机和量子通信不可或缺的基础理论。它的发展会促进物理学、化学、生物学等自然学科的交融,推动量子技术的进步。量子控制主要涵盖了量子态的操纵与制备、无退相干空间的构建等一系列基本课题。量子控制的研究进展也将有助于提升量子信息在未来通信中的地位。本文重点研究了Lyapunov控制在不同量子系统中若干应用,包括自旋-1/2链、无自旋费米子1维Kitaev链、1维双势阱光晶格、拓扑超导线与量子点的混合系统以及囚禁于光晶格的1维费米气体。这些应用将会对未来的量子信息处理起到一定的促进作用。我们的研究工作从第三章开始。第一章和第二章阐述了研究工作的背景,并给出了Lyapunov急定性定理和不变集定理等相关的基本定理,为接下来的研究工作提供了理论基础。此外,还简要介绍了Lyapunov函数几种设计方法。第三章中,通过调节边界自旋与近邻自旋的耦合强度或边界自旋的Larmor频率,实现自旋-1/2链中量子态的高保真度转移。与传统量子态转移方法不同,该方法具有以下优点：系统末态是稳态；不需要精确控制系统末态的转移时间；对控制场的扰动具有很好的鲁棒性。该方法还可以被应用到不同周期结构自旋链的量子态转移。第四章中,在二次型哈密顿量所描述的量子系统中,实现将准粒子驱动到拓扑模(topological modes)。在费米系统中,以无自旋费米子1维Kitaev链为例展示如何获取Majorana零模。在玻色系统中,以Su-Schrieffer-Heeger模型为例展示如何将系统驱动到边界模式(edge mode)。最后探讨将含时控制场波形替换成方波脉冲的可能性。第五章中,借助于拓扑超导线中的Majorana费米子,采用四种不同方案实现两量子点的长程纠缠。即,隐形传态方案；交叉Andreev反射方案；内点自旋翻转方案；超越内点自旋翻转方案。我们分别利用Lyapunov控制和绝热过程来形成长程纠缠。相比于绝热过程,Lyapunov控制的优势体现在灵活选择控制哈密顿量和缩短控制时间上。第六章中,研究如何在Aubry-Andre-Harper模型中实现边界态(edge state)。该方案的优势体现在只需调节边界格点的能量就能实现边界态。然后我们利用变形的Lyapunov函数来设计控制场实现边界-边界纠缠,也就是,两个边界态的最大纠缠态。该方法为边界态的操纵提供了一种新的有效手段。最后一章给出了本文的总结和展望。
[Abstract]:As a new interdisciplinary subject, quantum cybernetics is an indispensable basic theory to realize quantum computer and quantum communication. Its development will promote physics, chemistry, biology and other natural disciplines, and promote the progress of quantum technology. Quantum control mainly covers a series of basic issues, such as the manipulation and preparation of quantum states, the construction of non-decoherence space and so on. The research progress of quantum control will also help to enhance the status of quantum information in future communications. In this paper, we focus on the applications of Lyapunov control in different quantum systems, including spin 1 / 2 chain, spin fermion free 1 dimensional Kitaev chain, 1 dimensional double potential well optical lattice, and 1 D double potential well optical lattice. A hybrid system of topological superconducting wires and quantum dots and 1 D Fermi gas trapped in optical lattices. These applications will promote quantum information processing in the future. Our research begins with chapter three. In the first and second chapters, the background of the research work is described, and some basic theorems, such as Lyapunov's theorem of urgency and invariant set theorem, are given, which provide the theoretical basis for the next research work. In addition, several design methods of Lyapunov function are briefly introduced. In chapter 3, the high fidelity transfer of quantum states in the spin 1 / 2 chain is realized by adjusting the coupling strength between the boundary spin and the nearest neighbor spin or the Larmor frequency of the boundary spin. Different from the traditional quantum state transfer method, this method has the following advantages: the final state of the system is steady-state, the transition time of the final state of the system does not need to be controlled accurately, and it is robust to the disturbance of the control field. The method can also be applied to the quantum state transfer of spin chains with different periodic structures. In chapter 4, in the quantum system described by the quadratic Hamiltonian, the quasi-particle is driven to the topological mode (topological modes). In a Fermi system, a one-dimensional Kitaev chain without spin fermions is taken as an example to show how to obtain Majorana zero modules. In Bose systems, take the Su-Schrieffer-Heeger model as an example to show how to drive the system to the boundary mode (edge mode). Finally, the possibility of replacing time-dependent control field waveform with square wave pulse is discussed. In chapter 5, by means of the Majorana fermion in the topological superconducting wire, four different schemes are used to realize the long-range entanglement of two quantum dots. That is, the teleportation scheme; the cross Andreev reflection scheme; the internal point spin flip scheme; the transcendental internal point spin flip scheme. We use Lyapunov control and adiabatic process to form long-range entanglement respectively. Compared with adiabatic process, the advantage of Lyapunov control lies in flexible control of Hamiltonian and shortening of control time. In the sixth chapter, we study how to realize the boundary state (edge state). In the Aubry-Andre-Harper model. The advantage of this scheme is that the boundary state can be realized only by adjusting the energy of the boundary lattice. Then we use the deformed Lyapunov function to design the control field to realize the boundary-boundary entanglement, that is, the maximal entangled state of two boundary states. This method provides a new effective method for boundary state manipulation. The last chapter gives the summary and prospect of this paper.
【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O413;O231

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