Implicit Lyapunov Control for the Quantum Liouville Equation
For researchers in quantum control, this work solves a known convergence issue in degenerate quantum systems, but the method is incremental as it builds on existing Lyapunov control techniques.
The paper addresses convergence problems in multi-control Hamiltonians closed quantum systems under degenerate conditions by introducing implicit function perturbations and an implicit Lyapunov function. The proposed method ensures convergence from any initial state to an arbitrary target state unitarily equivalent to the initial state, with numerical simulations verifying effectiveness.
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. 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 tar-get 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. Finally, numerical simulations are studied to verify the effectiveness of the proposed control method.