Pulse design for multilevel systems by utilizing Lie transforms

We put forward a scheme to design pulses to manipulate multilevel systems with Lie transforms. A formula to reverse construct a control Hamiltonian is given and is applied in pulse design in the three- and four-level systems as examples. To demonstrate the validity of the scheme, we perform numerical simulations, which show the population transfers for cascaded three-level and N-type four-level Rydberg atoms can be completed successfully with high fidelities. Therefore, the scheme may benefit quantum information tasks based on multilevel systems.