Modeling And Control Of Open Quantum Systems

Open quantum systems are common in universe. How to apply the control theory to open quantum systems to accomplish varieties of control tasks under the impact of the environments plays an important role in the estabilishment of quantum control theory. In this thesis, open quantum systems are the objects of study. The main contents include modeling of open quantum systems, population transfer control, state transfer control and maintaining of coherence based on the system's model. The following 4 aspects are specific contents:1. According to the two categories of the baths with energy conservation or not, the Markovian master equations of open quantum systems are established and the building process is analyzed. In the category of bath with energy conservation, the Born master equation is derived by means of the Born approximation, and the Markovian master equation about the reduced density matrix of an open quantum system is derived by the Markov approximation further. The Lindblad form of master equation, the GKS expression, the Bloch sphere expression and the Kraus expression are the deformations of the Markovian master equation. Four kinds of Markov approximations that can remove the non-Markovian property are compared. The post-Markovian master equation that can meet the perfect requirements of dissipative dynamic when the Markov approximations are not suitable to the open quantum system is analyzed. When energy exchanges between the heat bath and the quantum system, while the total system composed of the bath and the quantum system has a conservation of energy, the dynamic equation of the open quantum system is presented, and the corresponding Markovian master equation is derived by the Markov approximation.2. According to the dissipative dynamics of open quantum systems, an optimal control method without iteration is proposed to transfer the population and the state. The decision problem of the shortest path for population transfer of quantum systems is solved by the Markov process. According to the Lindblad form of open quantum systems with relaxation and dissipation, the dynamic differential equation in matrix form is transformed into a vector form by Liouville super-operator. Different performance index are chosen, and two optimal control laws without iterations are designed to achieve the population transfer and the state transfer from an arbitrary initial state to an arbitrary target state under a particular condition. 3. Based on the decoherence-free subspaces (DFS), a dual control scheme is proposed for state transferring and coherence maintaining of open quantum systems: regarding aΛ-type N level open quantum system, an external laser field I is designed to construct the expected target state to be a decoherence-free state in DFS, which is decoupling from the environment; a control fieldⅡis designed by Lyapunov control method to transfer an anbitrary initial state to the expected target state, while the coherence of the state being maintained. The idea of designing these two control fields is extended to N level open quantum systems with general structure. The largest invariant set is shrinked by designing control Hamiltonians. With the Lyapunov control law, the excited state can be transferred to the superposition states in DFS and stay there, thereby the coherence of the system state can be maintained.4. Research on the performance comparison based on simulations of quantum system control. The difference between the expectation of the target state in the controlled state and control energy consumed was selected as the performance index, an optimal control law for closed quantum systems was derived. A Lyapunov-based control law was derived by constructing the sum of the average value of an imaginary mechanical quantity and control energy as the Lyapunov function. The effect of energy restriction and the relations between the Lyapunov control law and the optimal control law are discussed. Finally, the effects caused by the choice of parameters and control strategies were compared by means of the series of simulation experiments on a spin-1/2 particle system and a five-level quantum system.