Protecting coherence in optimal control theory: State-dependent constraint approach

Optimal control theory is developed for the task of obtaining an objective in a subspace of the Hilbert space while avoiding population transfer to other subspaces. The objective, a state-to-state transition or a unitary transformation, is carried out without loss of coherence, provided the system in the allowed subspace is decoupled from its environment. An optimization functional is introduced that leads to monotonic convergence of the algorithm. This approach becomes necessary for molecular systems which are subject to processes implying loss of coherence such as ionization or predissociation. In the subspaces corresponding to lossy channels, controllability is hampered or even completely lost. A functional constraint that depends on the state of the system at each instant in time keeps the system out of the lossy channels. We outline the resulting algorithm and discuss its convergence properties. The functionality of the algorithm is demonstrated for the examples of a state-to-state transition and of a unitary transformation for a model of cold Rb₂.