Research On The State Control And Coherence Preservation Of Quantum Systems

With the development of quantum technologies, the research domain of control science has been extended to the micro-world, and quantum control is becoming an important branch of control theory. The state control and coherence preservation of quantum systems are essensial in quantum control. Therefore, this thesis studies the state control based on Lyapunov theory and optimal control technique, and coherence preservation of open quantum systems. The main contents are as follows:1. The state control of quantum systems based on Lyapunov method is investigated. The first task is entanglement generation in a two-spin system. The mathematical model of a two-spin system under Heisengerg interaction is constructed in interaction picture. An average value of observable quantity is selected as the Lyapunov function. The observable operator is designed in order to make the target state be stable in Lyapunov sense. Then the convergence of the system controlled is analyzed, and the four Bell states are demonstrated to be global asymptotically stable under the control law designed. Secondly, for the purification of a qubit system, an auxiliary is introduced, whose interaction with the origin system is designed by the Lyapunov method so that the state evolution of the origin systm is non-unitary, as a result, any mixed sate of the origin system can be purified to an eigenstate. The last one is state control of open quantum systems:The control laws are designed to drive the system to a pure target state in decoherence-free subspace (DFS) in the interaction picture, where the average of an observable operator P is chosen as a Lyapunov function. By analyzing the positive limit set via Barbalat’s lemma, a sufficient condition on the observable operator is proposed such that the positive limit set only contains the target state. Under this condition, it is proved that the control laws can drive the open system to any pure target state in DFS and a procedure to construct the observable operator via Schmidt orthogonalization is given.2. The optimal control of state-to-state transiton probability is explored in the framework of quantum control landscape. For n-level open quantum systems governed by the Lindblad equation, the coherence vector representation of density matrix is introduced to transform the master equation to a standard bilinear controlled equation on the real space. For this system, the state-to-state transiton probability is defined as the control object, which is a functional of control functions, is referred to as the quantum control landscape. The numerical method using DMORPH algorithem is presented to explore the critical points of state-to-state transition probability control landscape in the control space. For a two level system, it is shown analytically that the control landscape corresponding to the transition probability from the North pole to a target on the Bloch sphere generally does not possess any control satisfying the critical point condition. The numerical results demonstrate that a finite optimal target time exists such that the corresponding transition probability reaches its highest value. Each optimal control at the optimal target time contains temporal short sub-pulses, similar to that for the time optimal control of analogous closed quantum systems with unbounded controls.3. Coherence preservation of high dimensional quantum systems subject to Markov decoherence is studied. First, a A-type three-level atom model is considered. The coherence preservation between a ground state and the excited state is defined as the control object, and a control field is designed to keep the coherence constant. For the possible singularities of the control field, we qualitatively analyze the breakdown time, i.e. the time of control diverging. In the case that one decay rate is considerably larger than the other one, the region in which the state stays to maintain coherence within a long time is obtained. For other cases, the influence of various parameters on the breakdown time is investigated, the results are verified by the simulation experiments. Secondly, Coherence preservation of E-type N-level atoms is investigated. The coherence is embodied by the coherence functions that are defined in an N-dimension Hilbert space. To design external control field that maintains the coherence functions at a constant value, the number of components of the field should be the same as the number of coherence functions. And each component addresses a particular coherence function. In addition, the parameters affecting singular points are analyzed and the possible way to extend the coherence preservation is introduced.