量子隐李雅普诺夫控制方法及相关应用研究
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摘要
完善的量子控制理论对于量子计算、原子物理、选键化学、量子光学、纳米材料、量子通信等领域的飞速发展具有重要的意义。在众多量子控制方法中,基于李雅普诺夫稳定性理论的量子控制方法的设计较简单,可以得到解析的控制律,且基于该控制方法的控制系统至少是稳定的,所以是一种比较常用的量子控制方法。在量子系统的特性及其控制的研究中,封闭量子系统的特性及其控制的研究是开放量子系统的特性及其控制的研究基础。从控制的角度看,对封闭量子系统的控制主要包括状态转移和轨迹跟踪两个方面。目前,为实现封闭量子系统的状态转移所采用的量子李雅普诺夫控制方法的研究主要集中在处于退化情况的量子系统。然而在实际中,很少有量子系统处于非退化情况,而大多数量子系统是处于退化情况。所以本论文主要研究的是采用隐李雅普诺夫控制方法来解决封闭量子系统退化情况下的任意两个态之间的完全状态转移控制问题,其中包括任意的本征态、叠加态和混合态。并研究量子控制在实际中的一个应用：相干反斯托克斯拉曼散射(CARS)相邻能级的选择激发。
首先,对于薛定谔方程,通过在控制律中引入隐函数微扰控制项和分别选取基于状态间距离、状态间偏差和虚拟力学量均值的隐函数李雅普诺夫函数解决了控制系统退化情况下从任意纯态初始态到任意本征态目标态的完全转移,也就是收敛性问题。又通过在控制律中引入常值扰动控制项解决了目标态为叠加态时的收敛性问题。并根据拉萨尔不变原理分别证明了在三种隐李雅普诺夫控制方法下控制系统的收敛性。分析了如何使收敛条件成立。通过数值仿真实验来验证所提出控制方法的正确性和有效性。并分析了三种隐函数李雅普诺夫函数的关系和三种隐李雅普诺夫控制方法的优缺点,对比三种控制方法的控制效果。
第二,对于刘维尔方程,通过在控制律中引入隐函数微扰控制项,并选取基于虚拟力学量均值的隐函数李雅普诺夫函数解决了控制系统退化情况下从任意初始态到与内部哈密顿对易的目标态的收敛性问题。又通过在控制律中引入常值扰动控制项,解决了目标态为非对易与内部哈密顿的态时的收敛性问题。根据拉萨尔不变原理证明了基于该控制方法的控制系统的收敛性。分析了如何使收敛性条件得到满足,从而提出了便于设计的显式的虚拟力学量设计原则。通过几个数值仿真实验来验证所提出控制方法的正确性和有效性。
第三,在实际的应用中,飞秒CARS用于探测物质能谱的灵敏度较高,被广泛应用到许多领域。飞秒CARS相邻能级的选择激发是CARS的一个重要的研究方向。目前实现CARS相邻能级选择激发的方法主要集中在实验上,分析和总结较少。本论文采用Silberberg提出的控制方法,在一系列参数调整实验的基础上,总结了各可调参数对选择激发效果的影响。并根据相关理论,定性分析了该方法的内部控制机理,分析了最佳可调参数能够实现相邻能级选择激发的原因,以及最佳可调参数的大致范围。从而总结出了实现相邻能级选择激发的参数调控方法。
The establishment of the complete quantum control theory has the great significance for the rapid development of the quantum computing, atomic physics, bond-selective chemistry, quantum optics, nanomaterials, quantum communication and other fields. Among quantum control methods, the control laws design of the quantum control method based on the Lyapunov stability theory is relatively simple, and the obtained control laws are analytical. Moreover, the control system based on this control method is at least stable. Therefore the quantum Lyapunov control method is a common quantum control method. Research on the characteristics and the control of closed quantum systems is the basis of that of open quantum systems. The control of closed quantum systems mainly includes two aspects as the state transfer and the trajectory tracking. At present, the study of the quantum Lyapunov control method for the state transfer in closed quantum systems mainly focus on the non-degenerate cases. However, in practice, most of the practical quantum systems are in the degenerate cases but not the non-degenerate cases. This thesis mainly studies that the implicit Lyapunov control method to solve the control problem of the state transfer between two arbitrary states in the closed quantum systems which is in the degenerate cases.The so-called "arbitrary" means any eigenstate, superposition state or mixed state. And this thesis studies an application of the quantum control: the selective excitation of the coherent anti-Stokes Raman Scattering (CARS).
First of all, for the bilinear Schrodinger equation, by means of introducing a series of control items which are implicit function perturbations, and choosing the implicit Lyapunov functions based on the state distance, state error, and the average value of an imaginary mechanical quantity, respectively, the complete state transfer problem of the control system in the degenerate cases from an arbitrary intial pure state to an arbitrary target eigenstate, i.e., the convergence problem, is solved. By means of introducing a series of constant disturbance control items at the same time, the convergence for the case that the target state is a superposition state is solved. The convergence of the control system based on these three implicit Lyapunov control methods, respectively, is proved according to the LaSalle invariance principle. How to make the convergence conditions be satisfied is also analyzed in this thesis. Some numerical simulation experiments are done to verify the correctness and the effectiveness of these three proposed implicit Lyapunov control methods. Moreover, the relation among three implicit Lyapunov functions, and the advantages and disadvantages of these three implicit Lyapunov quantum control methods are analyzed. The control effects of these three control methods are compared.
Secondly, for the quantum Liouville equation, by means of introducing a series of control items which are implicit function perturbations, and choosing the implicit Lyapunov function based on the average value of an imaginary mechanical quantity, the convergence problem of the control system in the degenerate cases from an arbitrary intial state to the target state which commutes with the internal Hamiltonian is solved. By means of introducing a series of constant disturbance control items at the same time, the convergence for the case that the target state does not commute with the internal Hamiltonian is solved. The convergence of the control system based on this implicit Lyapunov control method is proved according to the LaSalle invariance principle. How to make the convergence conditions be satisfied is also analyzed in this thesis. Some numerical simulation experiments are done to verify the correctness and the effectiveness of the implicit Lyapunov control method based on the average value of an imaginary mechanical quantity proposed in this thesis.
Thirdly, in the application, the sensitivity of the detection for the material energy spectrum by using the femtosecond CARS is high. The femtosecond CARS has been widely applied in many fields. The selective excitation of femtosecond CARS is an important research focus of CARS. At present, the selective excitation of CARS mainly focuses on the experiment. However, there exists few analysis and summary in the study of the selective excitation of CARS. In this thesis, the method proposed by Silberberg is used. The effect of various adjustable parameters on the selective excitation is summarized by means of parameters adjustment experiments. And according to the relevant theory, the control mechanism of the control method, the reason of the selective excitation of CARS by using the best adjustable parameters, and the approximate range of the best adjustable parameters are qualitatively analyzed. At last, the method of the parameter adjustments for the selective excitation is summarized.
首先,对于薛定谔方程,通过在控制律中引入隐函数微扰控制项和分别选取基于状态间距离、状态间偏差和虚拟力学量均值的隐函数李雅普诺夫函数解决了控制系统退化情况下从任意纯态初始态到任意本征态目标态的完全转移,也就是收敛性问题。又通过在控制律中引入常值扰动控制项解决了目标态为叠加态时的收敛性问题。并根据拉萨尔不变原理分别证明了在三种隐李雅普诺夫控制方法下控制系统的收敛性。分析了如何使收敛条件成立。通过数值仿真实验来验证所提出控制方法的正确性和有效性。并分析了三种隐函数李雅普诺夫函数的关系和三种隐李雅普诺夫控制方法的优缺点,对比三种控制方法的控制效果。
第二,对于刘维尔方程,通过在控制律中引入隐函数微扰控制项,并选取基于虚拟力学量均值的隐函数李雅普诺夫函数解决了控制系统退化情况下从任意初始态到与内部哈密顿对易的目标态的收敛性问题。又通过在控制律中引入常值扰动控制项,解决了目标态为非对易与内部哈密顿的态时的收敛性问题。根据拉萨尔不变原理证明了基于该控制方法的控制系统的收敛性。分析了如何使收敛性条件得到满足,从而提出了便于设计的显式的虚拟力学量设计原则。通过几个数值仿真实验来验证所提出控制方法的正确性和有效性。
第三,在实际的应用中,飞秒CARS用于探测物质能谱的灵敏度较高,被广泛应用到许多领域。飞秒CARS相邻能级的选择激发是CARS的一个重要的研究方向。目前实现CARS相邻能级选择激发的方法主要集中在实验上,分析和总结较少。本论文采用Silberberg提出的控制方法,在一系列参数调整实验的基础上,总结了各可调参数对选择激发效果的影响。并根据相关理论,定性分析了该方法的内部控制机理,分析了最佳可调参数能够实现相邻能级选择激发的原因,以及最佳可调参数的大致范围。从而总结出了实现相邻能级选择激发的参数调控方法。
The establishment of the complete quantum control theory has the great significance for the rapid development of the quantum computing, atomic physics, bond-selective chemistry, quantum optics, nanomaterials, quantum communication and other fields. Among quantum control methods, the control laws design of the quantum control method based on the Lyapunov stability theory is relatively simple, and the obtained control laws are analytical. Moreover, the control system based on this control method is at least stable. Therefore the quantum Lyapunov control method is a common quantum control method. Research on the characteristics and the control of closed quantum systems is the basis of that of open quantum systems. The control of closed quantum systems mainly includes two aspects as the state transfer and the trajectory tracking. At present, the study of the quantum Lyapunov control method for the state transfer in closed quantum systems mainly focus on the non-degenerate cases. However, in practice, most of the practical quantum systems are in the degenerate cases but not the non-degenerate cases. This thesis mainly studies that the implicit Lyapunov control method to solve the control problem of the state transfer between two arbitrary states in the closed quantum systems which is in the degenerate cases.The so-called "arbitrary" means any eigenstate, superposition state or mixed state. And this thesis studies an application of the quantum control: the selective excitation of the coherent anti-Stokes Raman Scattering (CARS).
First of all, for the bilinear Schrodinger equation, by means of introducing a series of control items which are implicit function perturbations, and choosing the implicit Lyapunov functions based on the state distance, state error, and the average value of an imaginary mechanical quantity, respectively, the complete state transfer problem of the control system in the degenerate cases from an arbitrary intial pure state to an arbitrary target eigenstate, i.e., the convergence problem, is solved. By means of introducing a series of constant disturbance control items at the same time, the convergence for the case that the target state is a superposition state is solved. The convergence of the control system based on these three implicit Lyapunov control methods, respectively, is proved according to the LaSalle invariance principle. How to make the convergence conditions be satisfied is also analyzed in this thesis. Some numerical simulation experiments are done to verify the correctness and the effectiveness of these three proposed implicit Lyapunov control methods. Moreover, the relation among three implicit Lyapunov functions, and the advantages and disadvantages of these three implicit Lyapunov quantum control methods are analyzed. The control effects of these three control methods are compared.
Secondly, for the quantum Liouville equation, by means of introducing a series of control items which are implicit function perturbations, and choosing the implicit Lyapunov function based on the average value of an imaginary mechanical quantity, the convergence problem of the control system in the degenerate cases from an arbitrary intial state to the target state which commutes with the internal Hamiltonian is solved. By means of introducing a series of constant disturbance control items at the same time, the convergence for the case that the target state does not commute with the internal Hamiltonian is solved. The convergence of the control system based on this implicit Lyapunov control method is proved according to the LaSalle invariance principle. How to make the convergence conditions be satisfied is also analyzed in this thesis. Some numerical simulation experiments are done to verify the correctness and the effectiveness of the implicit Lyapunov control method based on the average value of an imaginary mechanical quantity proposed in this thesis.
Thirdly, in the application, the sensitivity of the detection for the material energy spectrum by using the femtosecond CARS is high. The femtosecond CARS has been widely applied in many fields. The selective excitation of femtosecond CARS is an important research focus of CARS. At present, the selective excitation of CARS mainly focuses on the experiment. However, there exists few analysis and summary in the study of the selective excitation of CARS. In this thesis, the method proposed by Silberberg is used. The effect of various adjustable parameters on the selective excitation is summarized by means of parameters adjustment experiments. And according to the relevant theory, the control mechanism of the control method, the reason of the selective excitation of CARS by using the best adjustable parameters, and the approximate range of the best adjustable parameters are qualitatively analyzed. At last, the method of the parameter adjustments for the selective excitation is summarized.
引文
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