Parametrizations of a two-level quantum control system

A known two-level population transfer model of a quantum system is studied. Construction of the basic four-dimensional real-variable model is repeated first. By using a technique of underdetermined systems of ordinary differential equations (ODE) the two scalar control variables are represented then as functions of the state variables. This representation is used to obtain an underdetermined system of nonlinear ODEs, which does not include the control variables. Via a flatness-based idea two state variables are used to parametrize the remaining ones and the two controls. The coordinate system of the states is converted into polar form. Then another form of parametrization is obtained for the states and controls. A quadratic energy minimizing optimization is also studied. It is converted via the two parametrizations to two equivalent variational problems. A general solution to the polar-form variational problem is given. Simulation results to be presented in a companion paper will complete results of this study.