Studies Of Quantum System Control Methods Based On Lyapunov And Optimal Control Theory

The establishment of complete quantum control theory is important for the development of the fields such as physics, chemistry, quantum information, nanotechnology and biotechnology. For the complexity and particularity of quantum systems themselves, research on quantum control needs a great deal of endeavor from the scholars in various fields. Under such a background, from the viewpoint of the control theory this thesis studies the applications of Lyapunov theory and optimal control theory to closed quantum systems and some aspects in open quantum systems, on the basis of pointing out the necessities of studying quantum control and reviewing the development status of the quantum control theory and its methods. The main contents are as follows.1) For the state control problem in closed quantum systems, according to the distance between states, the error between states and the average value of an imaginary mechanical quantity, three Lyapunov functions are chosen, respectively. The corresponding control law designs and the convergence analyses are studied. Especially, for the Lyapunov function based on the distance between states, a class of control laws that can solve the problem that an initial state is orthogonal to a goal state is designed. The convergence of the system is analyzed via LaSalle's principle and one method to find the largest invariant set of the closed loop system is proposed by linearizing the unitary evolution operator at the corresponding instants. For the Lyapunov function based on the average value of an imaginary mechanical quantity, the largest invariant set of the closed loop system under some weaker conditions is deduced. Then, the structure of the largest invariant set is analyzed and the construction method of the imaginary mechanical quantity is proposed. The validity of the three Lyapunov design methods is verified by numerical simulation experiments. The corresponding control effects are compared. The relations among the three Lyapunov functions are analyzed and one of their unified forms is given. The characteristics of the three Lyapunov methods are summarized.2) Taking the Lyapunov method based on the state distance as an example, this thesis proposes the vector design method of the control laws and extends the range of the control values.3) For the optimal control of closed quantum systems, the function state form of quantum states is utilized to deduce quantum optimal control equations. Based on the on-off variable metric method, an efficient numerical method with optimized search step length is proposed.4) The development status of quantum state estimation methods is reviewed. These estimation methods are classified by different measurement styles. The quantum trajectory theory under the influence of continuous measurements is introduced and the relation between the master equation followed by the average post-measurement state and different measurement schemes is pointed out.