  Adiabatic elimination is a standard tool in quantum optics, which produces an
effective Hamiltonian for a relevant subspace of states, incorporating effects
of its coupling to states with much higher unperturbed energy. It shares with
techniques from other fields the emphasis on the existence of widely separated
scales. Given this fact, the question arises whether it is feasible to improve
on the adiabatic approximation, similarly to some of those other approaches. A
number of authors have addressed the issue from the quantum optics/atomic
physics perspective, and have run into the issue of non-hermiticity of the
effective Hamiltonian improved beyond the adiabatic approximation, which poses
conceptual and practical problems. Here, we first briefly survey methods
present in the physics literature. Next, we rewrite the problems addressed by
the adiabatic elimination technique to make apparent the fact that they are
singular perturbation problems from the point of view of dynamical systems. We
apply the invariant manifold method for singular perturbation problems to this
case, and show that this method produces the equation named after Bloch in
nuclear physics. Given the wide separation of scales, it becomes intuitive that
the Bloch equation admits iterative/perturbative solutions. We show, using a
fixed point theorem, that indeed the iteration converges to a perturbative
solution that produces in turn an exact Hamiltonian for the relevant subspace.
We propose thus several sequences of effective Hamiltonians, starting with the
adiabatic elimination and improving on it. We show the origin of the
non-hermiticity, and that it is inessential given the isospectrality of the
effective non-hermitian operator and a corresponding effective hermitian
operator, which we build. We propose an application of the introduced
techniques to periodic Hamiltonians.
