Wei-Norman technique for control design of bilinear ODE systems with application to quantum control

Abstract

A two-level quantum system model describing population transfer driven by a laser field is studied. A four-dimensional real-variable differential equation model is first obtained from the complex-valued two-level model describing the wave function of the system. Due to bilinearity in the control and the states Lie-algebraic techniques can be applied for constructing the state transition matrix of the system. The Wei-Norman technique is used in the construction. The exponential representation of the transition matrix includes three base functions, two of which serves as the parameter functions, which can be chosen freely. This corresponds to considering the overall control system as an underdetermined differential system. In this framework the initial and final states can be defined corresponding to the two levels of the original system model. Then flatness-based design is applied for explicitly calculating the parameter functions, which in turn give the desired input-output pairs. This input then drives the state of the system from the given initial state to the given final state in a finite time. © 2010 Springer-Verlag Berlin Heidelberg.