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The convergence of closed quantum systems in the degenerate cases to thedesired target state by using the quantum Lyapunov control based on the averagevalue of an imaginary mechanical quantity is studied. On the basis of theexisting methods which can only ensure the single-control Hamiltonian systemsconverge toward a set, we design the control laws to make the multi-controlHamiltonian systems converge to the desired target state. The convergence ofthe control system is proved, and the convergence to the desired target stateis analyzed. How to make these conditions of convergence to the target state tobe satisfied is proved or analyzed. Finally, numerical simulations for a threelevel system in the degenrate case transfering form an initial eigenstate to atarget superposition state are studied to verify the effectiveness of theproposed control method.

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