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. Keywords: perturbations, Lyapunov control, degenerate, convergence, non-diagonal target stat

Model reduction of controlled Fokker--Planck and Liouville-von Neumann equations

Model reduction methods for bilinear control systems are compared by means of practical examples of Liouville-von Neumann and Fokker--Planck type. Methods based on balancing generalized system Gramians and on minimizing an H2-type cost functional are considered. The focus is on the numerical implementation and a thorough comparison of the methods. Structure and stability preservation are investigated, and the competitiveness of the approaches is shown for practically relevant, large-scale examples

A Survey of Quantum Lyapunov Control Methods

The condition of a quantum Lyapunov-based control which can be well used in a closed quantum system is that the method can make the system convergent but not just stable. In the convergence study of the quantum Lyapunov control, two situations are classified: non-degenerate cases and degenerate cases. In this paper, for these two situations, respectively, the target state is divided into four categories: eigenstate, the mixed state which commutes with the internal Hamiltonian, the superposition state, and the mixed state which does not commute with the internal Hamiltonian state. For these four categories, the quantum Lyapunov control methods for the closed quantum systems are summarized and analyzed. Especially, the convergence of the control system to the different target states is reviewed, and how to make the convergence conditions be satisfied is summarized and analyzed.Comment: 1