State Tracking And Operator Preparation In Quantum Systems

Quantum systems are grouped into closed systems and open systems according to whether the systems interact with their environment. The dissertation researches state control and operator manipulation in quantum systems. In closed systems, Lyapunov control laws for state transferring and state tracking are designed. Besides, a detailed convergence proof is accomplished. In open systems, the work focuses on complete state transferring and quantum gates operator implementation.In closed quantum systems, state transferring and state tracking are considered. Firstly, state transferring is theoretically realized in liquid nuclear magnetic resonance (NMR) system. In this example, another quantum system, the so-called auxiliary system, is coupled to NMR. Lyapunov control law is designed to transfer NMR from a high-mixed initial thermal equilibrium state to an effective pure state (pseudo-pure state). Secondly, state tracking methods are discussed in detail. Error as a new controlled variable is introduced to change state tracking into state transferring. If the system’s nature trajectory is tracked, interaction picture is also employed to change tracking into transferring. In this case, both the two methods produce the same control law if we choose the same Lyapunov function. Simulation experiments demonstrate that the tracking performance is related to target state and system structure. In the convergence proof, an ideal system, where energy differences between arbitrary levels of free Hamiltonian are different and the energy level is connected to each other, is investigated in interaction picture. To track the nature trajectory of the ideal controlled system, an average value of virtual observable quantity P is selected as Lyapunov function. The positive limit set of the controlled system is analyzed via Barbalat lemma. In the design process, P with non-degenerate eigen-spectrum is designed to shrink the limit set R to a smaller one, in which all the states in R commute with P; the eigenvalues of P is carefully selected to ensure convergence. To simplify the qualified P, the dissertation gives a candidate P, which is proved that the other states except the target one in the limit set under this P are critically stable. It is verified that a non-ideal system still converges to an eigen-state or pseudo-pure state described by diagonal density matrix, the sufficient and necessary condition of which is obtained.The open quantum systems interacted with Markovian and Non-Markovian environment are investigated. The dissipative dynamics in open quantum system prohibits the complete state tracking. In a four-level Markovian system, decoherence-free subspace is employed to track a nature trajectory of a decoherence-free target state. That this non-ideal system under the designed control law converges to target state is verified by theoretical description and simulation experiments.The Lyapunov control law is simplified to avoid oscillating state trajectory in a two-level Non-Markovian quantum system. Although a complete tracking is impossible except the case of equilibrium state tracking, a discrete target trajectory composed of pure states is implemented by path planning in simulation experiments.The single quantum gates in Non-Markovian system is manipulated by Lyapunov control. The singularity problem arising in control law is studied and solved. A redesigned Lyapunov control law brings an great improvement on control effectiveness:1) it ensures that the system reach target operator;2) the operator preparation process is finished in a short time because of the monotonically decreasing of Lyapunov function. Compare with optimal control, the fast preparation process is obvious and the adaptive control law has a peculiar advantage that it keeps target operator unchanged for a significant time. Moreover, the optimal accuracy is also obtained by a proper Lyapunov function. In the simulation experiments, another benefit of the proposed control law is the nice robustness against the variations of system parameters.