开放量子系统动力学不变量及其应用

Lewis-Riesenfeld dynamical invariant theory of closed quantum systems has become a hot topic in the study of physics, which has been applied to solve a time-dependent Schrodinger equation, to control a quantum state, and to track quantum trajectory of a quantum system. In recent works, we have successfully generalized the definition of Lewis-Riesenfeld dynamical invariants into open quantum systems. Further studies on properties and applications of the dynamical invariants of open quantum systems are imperative to put use in investigating the quantum information theory and the fundamental conception of quantum theory. In this project, we will carefully analyze the properties of the dynamical invariants of open quantum systems, and apply them in the fields of the quantum information and the quantum control theory as well. We will be focusing on resolving the following questions in this project: (1) an universal scheme to obtain the dynamical invariant will be developed, and a novel method in decomposing dynamical invariants are also designed to apply this theory into the quantum control scheme and the quantum information process; (2) the dynamical symmetry of open quantum systems will be analyzed based on the dynamical invariant theory. In the field of the application of the dynamical invariant theory of open quantum systems, we will pay our attention to the following aspects: (1) a kind of dynamical invariant with special structure will be designed, which satisfies the von Neumann equation. By using this dynamical invariant, the inverse engineering scheme will be developed to realize a rapid quantum information process without any effect of deocherence. Moreover, we also design another kind of inverse engineering scheme, in which the quantum state we operate does not have to be pure state. (2) Based on the dynamical invariant theory of open quantum systems, we will also exploit a general method to solve the dynamical equation of open quantum systems.

封闭量子系统的动力学不变量理论可用来求解薛定谔方程、设计量子控制方案、快速实现轨迹跟踪（trajectory tracking）。它是近几年来物理学研究领域的热门问题之一。在近期的工作中，申请人已成功地将不变量的定义推广至开放量子系统，但是对于开放量子系统不变量的性质和应用仍需深入的探索。本立项的目的就是要深入剖析开放量子系统的不变量理论，并将其应用到量子信息和量子控制领域，特别是开放系统的量子信息处理和开放系统的量子控制理论。本立项将解决如下问题：（1）研究动力学不变量构建的通用方法，并给出可用于设计控制方案和求解动力学方程的不变量分解方式；（2）运用不变量理论深入分析开放系统对称性。在应用方面，本立项解决如下问题：（1）设计开放系统中的Inverse Engineering控制方案，实现快速的、容错的量子信息处理过程，并研究相对应的绝热定理；（2）给出求解开放系统动力学方程的一般方法。

封闭量子系统的动力学不变量被定义为一个拥有常数期望值的含时厄米算符，该理论最初被设计用来研究存在含时哈密顿量的动力学演化问题。近年来，该理论在量子绝热捷径控制中的应用使动力学不变量理论再次得到了广泛的关注。然而，对此理论的研究一直局限于封闭量子系统，对于开放量子系统中的动力学不变量理论及其应用的研究仍属空白。本项目的目标就是研究分析开放系统动力学不变量的性质，并将其应用于开放量子系统的控制，从而实现适用于开放量子系统的绝热捷径技术。在本项目执行期间，我们总体上按照原定计划开展研究。同时，根据本研究领域的前沿发展动态我们对研究内容适当地进行了微小调整。我们主要采用量子开放系统领域中传统的解析推导与数值计算相结合的方法，研究了如何通过对称性构建量子开放系统的动力学不变量，如何利用量子开放系统动力学不变量实现量子开放系统的绝热捷径，以及如何将此绝热捷径技术应用于量子信息过程和量子热力学过程等物理问题，获得了一系列创新性成果。本项目的研究实现对量子开放系统中量子态的操控、量子相干器件的设计具有重要理论意义；对量子开放多体系统和量子热机的物理过程的理解以及量子信息处理技术和高精密测量技术的发展具有积极的推动作用。在项目实施的三年中（2017~2019），共发表由基金资助的SCI论文8篇。

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