Quantum control design by Lyapunov trajectory tracking for dipole and polarizability coupling

In this paper we analyze the Lyapunov trajectory tracking of the Schroedinger equation for a coupling control operator containing both a linear (dipole) and a quadratic (polarizability) term. We show numerically that the contribution of the quadratic part cannot be exploited by standard trajectory tracking tools and propose two improvements: discontinuous feedback and periodic (time-dependent) feedback. For both cases we present theoretical results and support them by numerical illustrations.