拉普拉斯变换在开放量子系统力学中的应用

发布时间：2018-01-13 10:08

本文关键词：拉普拉斯变换在开放量子系统力学中的应用　出处：《吉林大学》2017年硕士论文　论文类型：学位论文

【摘要】：多体开放量子系统的动力学问题是量子信息科学的一个研究热点。一个开放量子系统的环境是由无穷多个自由度组成的热库,要精确求解这个系统的动力学就需要解决含有无穷变量的动力学方程。如果我们将量子系统与其周围环境看成一个大的闭合系统,采用薛定谔方程解析其动力学,这种方法理论上似乎可行。但是多体开放量子系统薛定谔方程的求解在现有计算能力下几乎是办不到的。因此对于开放量子系统而言,仅有薛定谔方程本身是远远不够的。由于开放量子系统不可避免地会与周围环境自由度发生相互作用,从而导致量子系统的一部分信息和能量流向环境。在最初的研究中,人们考虑环境自由度非常大,系统和环境之间的耦合非常弱的情况。在这种情况下可以采用玻恩-马尔可夫近似,系统动力学展现出马尔可夫特性。但是当系统与环境之间的耦合非常强时,开放量子系统的动力学是非马尔可夫性的。它流入环境的信息将在未来的某一时刻重新对系统造成影响。本论文在处理马尔可夫与非马尔可夫这两种动力学时,介绍了量子马尔可夫主方程和非马尔可夫量子态扩散方程。由于环境自由度的影响,必然导致量子系统的退相干,这给操控相干量子态带来很大困难。目前已有多种方法抑制退相干效应,本论文介绍动力学解耦调控法,即通过引入脉冲使得系统与环境部分解耦。开放量子系统动力学的研究模型在数学处理和物理实现上都是相当复杂的。因此我们需要通过各种数值计算或近似技术实现对不相关变量的约化,进而在约化态空间形成简单的描述。一般而言,精确解是探究物理模型的基础,物理模型一旦被精确地解析,它被应用到实验以及其他科研领域的可能性就会很大,因此我们更希望得到系统动力学的精确解而不是近似解,此时就要求我们精确求解系统动力学的积分微分方程。数学上主要用拉普拉斯变换的方法来求解积分微分方程,本论文主要采用拉普拉斯变换精确求解四个常用开放量子系统模型的动力学方程:(1)与两个环境相互作用的三能级原子模型;(2)耗散的JC模型;(3)囚禁在耗散腔里的两量子比特模型;(4)N量子比特与零温热库相互作用模型。这四个物理模型的精确求解表明了拉普拉斯变换法在开放量子系统动力学中的重要地位,也为将来处理更复杂的动力学方程提供理论依据和近似求解的基础。
[Abstract]:The dynamics of multi-body open quantum systems is a hot topic in quantum information science. The environment of an open quantum system is a heat pool composed of infinite degrees of freedom. In order to solve the dynamics of the system accurately, we need to solve the dynamics equation with infinite variables. If we think of the quantum system and its surrounding environment as a large closed system. The Schrodinger equation is used to analyze its dynamics. This method seems to be feasible in theory, but the solution of Schrodinger equation for multibody open quantum system is almost impossible under the existing computational power, so for the open quantum system. The Schrodinger equation alone is far from enough because open quantum systems inevitably interact with the degree of freedom of the surrounding environment. As a result, part of the information and energy of the quantum system flow to the environment. In the initial study, people considered the environment with great degree of freedom. The coupling between the system and the environment is very weak. In this case, the Bosn-Markov approximation can be used, and the dynamics of the system shows the Markov characteristic. But when the coupling between the system and the environment is very strong, the coupling between the system and the environment is very strong. The dynamics of an open quantum system is non-Markov. The information that flows into the environment will re-influence the system at some point in the future. This paper deals with both Markov and non-Markov dynamics. The quantum Markov master equation and the non-Markov quantum state diffusion equation are introduced. Due to the influence of the degree of freedom in the environment, the decoherence of the quantum system is inevitable. This makes it difficult to manipulate coherent quantum states. At present, there are many methods to suppress the decoherence effect. In this paper, the dynamic decoupling control method is introduced. The open quantum system dynamics model is very complicated in mathematical processing and physical implementation. Therefore, we need to use various numerical calculations or approximate techniques. The reduction of uncorrelated variables was achieved by surgery. In general, the exact solution is the basis of exploring the physical model, once the physical model is analyzed accurately. It is very likely that it will be applied to experiments and other fields of scientific research, so we prefer to obtain the exact solution of system dynamics rather than the approximate solution. At this point, we are required to solve the integro-differential equations of system dynamics accurately. In mathematics, the Laplace transformation is mainly used to solve the integrodifferential equations. In this paper, Laplace transform is used to solve the three-level atomic model of interaction between four open quantum system models and two environments. (2) dissipative JC model; (3) two quantum bit models trapped in a dissipative cavity; The exact solution of the four physical models shows the importance of the Laplace transform method in the dynamics of open quantum systems. It also provides the theoretical basis and approximate solution basis for the more complex dynamic equations in the future.
【学位授予单位】：吉林大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O413

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