Implicit Lyapunov Control for the Quantum Liouville Equation

A quantum system whose internal Hamiltonian is not strongly regular or/and control Hamiltonians are not full connected, are thought to be in the degenerate cases. The most actual quantum systems are in these degenerate cases. In this paper, convergence problems of the multi-control Hamiltonians closed quantum systems in the degenerate cases are solved by introducing implicit function perturbations and choosing an implicit Lyapunov function based on the average value of an imaginary mechanical quantity. For the diagonal and non-diagonal target states, respectively, control laws are designed. The convergence of the control system is proved, and an explicit design principle of the imaginary mechanical quantity is proposed. By using the proposed method, the multi-control Hamiltonians closed quantum systems in the degenerate cases can converge from any initial state to an arbitrary target state unitarily equivalent to the initial state in most cases. Finally, numerical simulations are studied to verify the effectiveness of the proposed control method. The problem solved in this paper about the state transfer from any initial state to arbitrary target state of the quantum systems in degenerate cases approaches a big step to the control of actual systems.