For quantum state trajectory tracking of density matrix in Liouville equation of quantum systems,with the help of concept in quantum system control,one can apply unitary transformation both to controlled system and free-evolutionary target system such as to change the time-variant and non-stationary target system into a stationary state.Therefore,the quantum state trajectory tracking problem becomes a steering one.State steering control law of the system transformed is designed by means of the Lyapunov stability theorem.Finally,numerical simulation experiments are given for a five-level energy quantum system.The comparison analysis of original system's trajectory tracking with other method illustrates the advantage in control time of the method proposed.
We propose a Lyapunov-based control approach for state transfer based on the decoherence-free target state. The expected target state is constructed to be a decoherence-free state in a decoherence-free subspace （DFS） by an external laser field I, so that the system state can be decoupled from the environment, and no more decoherence process will occur. With the decoherence-free target state, we design a Lyapunov-based control field II to steer the given initial state to the decoherence-free state of open quantum systems as completely as possible, and decouple the system state from the environment at the same time. In the end, it is verified that the state transfer control designed comes true on a A-type four-level atomic system, and the system can stay on the decoherence-free target state without coupling to environment.
