State transfer for closed quantum systems with dipole and polarizing coupling via implicit Lyapunov control

In this paper, the state transfer problem of finite dimensional closed quantum systems with dipole and polarizability coupling in non-ideal cases is investigated. The non-ideal cases are considered where the internal Hamiltonian of the system is not strongly regular [1, 2] and not all the eigenvectors of the internal Hamiltonian are directly coupled with the target state. Both cases are closely related to practical systems such as the one-dimensional oscillator. A class of Lyapunov functions are first defined by implicit functions and their existences are guaranteed by a fixed point theorem. Then, the local convergence analysis of the controlled system is investigated by the LaSalle invariance principle. Finally, the characterization of the largest invariant set is presented. Moreover, simulation studies are discussed to show the effectiveness and advantage of proposed methods.