State-Selective Excitation of Quantum Systems via Geometrical Optimization

We lay out the foundations of a general method of quantum control via geometrical optimization. We apply the method to state-selective population transfer using ultrashort transform-limited pulses between manifolds of levels that may represent, e.g., state-selective transitions in molecules. Assuming that certain states can be prepared, we develop three implementations: (i) preoptimization, which implies engineering the initial state within the ground manifold or electronic state before the pulse is applied; (ii) postoptimization, which implies engineering the final state within the excited manifold or target electronic state, after the pulse; and (iii) double-time optimization, which uses both types of time-ordered manipulations. We apply the schemes to two important dynamical problems: To prepare arbitrary vibrational superposition states on the target electronic state and to select weakly coupled vibrational states. Whereas full population inversion between the electronic states only requires control at initial time in all of the ground vibrational levels, only very specific superposition states can be prepared with high fidelity by either pre- or postoptimization mechanisms. Full state-selective population inversion requires manipulating the vibrational coherences in the ground electronic state before the optical pulse is applied and in the excited electronic state afterward, but not during all times.