We show how the dynamics of a specific subset of states can be separated from the dynamic of the total quantum state via a time-dependent projector-based formalism of adiabatic elimination. Within our formalism, we assume an explicit time dependency in the coupling between both subsystems. Additionally, we do not assume that the elements of the Hamiltonian commute, as in matter-wave optics, this is not given in general. Here, the center-of-mass degrees of freedom frequently need to be taken into account. Assuming non-commutativity also enables the application of this formalism to matrices of arbitrary dimension and the treatment of the adiabatic elimination in high dimensional systems, where the elimination may be needed to reduce the overall complexity. In particular, we apply our formalism to typical mechanisms in matter-wave optics, Raman, and Bragg diffraction.
Böhringer et al. (Tue,) studied this question.