Optimal Control Of Quantum System Based On Lyapunov Stability Theorem

The establishment of quantum mechanics opens the new way of exploring the microcosm for humankind. Recent years, quantum mechanics has gone deep into various fields. Quantum system control appears as a new subject for the sake of operating increasingly complicated quantum phenomena so that making the microcosm better. Now it has become an international prosperous research fields and obtained many results. This work reviews the advancement and current research of controllability and controlled methods on the quantum systems, and focuses on the Lyapunov method and optimal method for the closed quantum systems. The main results are as following:(1) The unitary matrix was constructed by using eigenstates of the controlled quantum system so as to perform a coordinate rotation on the controlled system. In such a way the local phases of all states of the system were eliminated. Then the error between the controlled state and the target state after transformation was selected as the Lyapunov function, and the system control law was designed based on Lyapunov stability theorem with the pre-condition that the system is kept stable. The detailed design procedure of the controller needed for preparation of the system states and the comparison analysis of the results were given. Finally, the computer numerical simulations on a spin-1/2 particle system were implemented to prepare both the eigen and superposition target states with the proposed control method, and the relation between the system state evolution time and control value was analyzed with different parameters.(2) By means of the design method for bilinear systems in the macroscopic field, in association with the characteristic that the mathematical model of the pure state quantum system is bilinear, and according to the selected performance index of the controlled system, the optimal control strategy in quantum systems was proposed based on the Lyapunov stability theorem. The detailed design process of optimal control law was derived. Some numerical simulations were performed on a spin-1/2 particle system. Experimental results demonstrate the effectiveness of the method proposed. At the same time, the influence of different parameters on the system control effects is analyzed.(3) An optimal control strategy of mixed state steering in finite closed quantum systems is proposed in this paper. Two different situations are considered: one is the target state is in statistical incoherent mixtures of energy eigenstates in which the density matrices of the target states are diagonal. Another is not all of the off-diagonal elements in the density matrices of the target states are zeros, in which the trajectory tracking problem is translated into the state steering problem by introducing the unitary transformation with all energy eigenstates in the inner Hamiltonian of the system controlled. Based on the Lyapunov stability theorem the stable parameters may be selected and the optimality of the control law proposed is proven. Moreover, the numerical simulations are performed on the diatomic molecule described by the Morse oscillator model and the spin-1/2 system under the control law proposed, respectively. The system simulation results demonstrate that the control strategy is efficient.