Quantum Lyapunov Control Based on the Average Value of an Imaginary Mechanical Quantity

Abstract

The convergence of closed quantum systems in the degenerate cases to the arbitrary desired target state by using the quantum Lyapunov control based on the average value of an imaginary mechanical quantity is studied in this paper. On the basis of the existing methods which can only ensure the pure states and single-control Hamiltonian systems converge toward a set, we propose a control laws design to make the multi-control Hamiltonian systems, which can also converge from the arbitrary initial state to the arbitrary target state of the quantum systems whose internal Hamiltonian are not strongly regular or/and control Hamiltonians are not full connected. The degenerate cases' problems solved in this paper widely exist in the practical quantum systems, so it has the great significance in quantum systems control. People can make use of those conditions obtained to design a convergent controller for the quantum control system, which can instruct the experimental scientists to obtain high successful probability to the actual quantum systems control. This research work establishes a completed quantum Lyapunov control theory in closed quantum systems. The convergence of the control system is proved. How to make the convergence conditions to be satisfied is proved or analyzed. Finally, numerical simulations for a three level system in the degenrate case transfering from an initial eigenstate to a target superposition state are studied.