On Manipulation Of Quantum States And Its Optimization

With the rapid development of quantum technologies,the research domain of control science has been extended to the micro-world, and quantum control is becoming an important issue of control theory.Although more and more attentions have been paid on the control problem of quantum systems in recent years, and some theoretical and experimental progresses have been made in this domain, the quantum control research is just at the beginning and there are still many directions need to be explored. Under such a background, this thesis studies the manipulation and optimization problems of quantum states by using the optimal conrtol theory from the viewpoint of the control theory. The main contents are summarized as follows:1) We make full use of the geometrical parametrization of quantum states, and explore how to manipulate quantum states by one-rotation controls based on constructing the special control Hamiltonian to steer the quantum system to the target state. Then the characteristics of the dynamical trajectory of quantum states are discussed under different manipulation condition. In addition, we should point out that the restriction of the control magnitude can affect the trajectory based on one-tunable control Hamiltonian, and it can not produce any influences on the courses of two or three-tunable control Hamiltonians. Furthermore, when the controls are unbounded, the trajectory of two-tunable control Hamiltonian and the one based on one-tunable control Hamiltonian are belong to the same circle on Bloch sphere. And it should be emphasized that the trajectory of two-tunable control Hamiltonian must be the minor arc in this circle. For one-tunable control Hamiltonian, the trajectory is not always a guarantee of the minor arc. Since the trajectory based on three-tunable control Hamiltonian is along the geodesic curve that joins the initial state and the target state at all events, it is the shortest arc on Bloch sphere.2) A new kind of weighted time-energy performance is introduced to trade-off time and energy resource cost. Then we explore how to design control magnitude to minimize this performance on the basis of the accomplishment of the control object. Furthermore, we analyse the influence of different manipulation conditions on the performance. And we would like to point out that when more-tunable control Hamiltonian is available, the better performance could be got. One physical examples and three concrete numerical examples are given to illustrate the feasibility and efficiency of this approach.3) In this paper, we just concentrate on four tipical kind functions in the domain of the quantum control theory: Bang-Bang control, triangle function, sinusoidal function and polynomial function. Then the particular optimal controls of these four kinds of functions are presented in terms of the optimal control problem of quantum states, respectively. In addition, a comparison has been made among the manipulative capability of these functions.After the review of the current main developments of quantum control, we study the preceding fruits of the investigation, and introduce a new approach to manipulate single qubits from the initial state to the target one. Then we analyse the influence of the different manipulation conditions on the dynamics of quantum systems. Moreover, a comparison has been made among the capability of these four kinds of functions. The main work and contributions of this thesis can promote the studies of the control problem of quantum systems.