Studies Of Implicit Lyapunov-Based Quantum Control And Relevant Applications

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.