Adiabatic elimination is the process of truncating a Hamiltonian's Hilbert space to the "slow" states you care about. You throw out the "fast" eigenstates to produce a smaller effective Hamiltonian $H_{eff}$ at the cost of having different couplings than the bare $H$.
My question is about the practical implementation: how does one adiabatically eliminate states to obtain $H_{eff}$ when given a Hamiltonian $H$? I would ideally like a formula that I could use on a computer using standard matrix operations
Following this paper by Sanz etc al https://arxiv.org/abs/1503.01369, they provide the following formula for $H_{eff}$ (equation 3)
$$H_{eff}=PHP-PHQ\frac{1}{QHQ}QHP$$
Where $P$ is the projector onto the slow eigenstates and $Q$ projects onto the fast ones. The above form looks very similar to what you get from perturbation theory, which I do not fully grasp because adiabatic elimination is presented as a non-perturbative method in textbooks.
I tried to apply this to the canonical example of a three-level lambda system with a Raman transition:
$$H= \begin{bmatrix} 0 & 0 & \Omega_P \\ 0 & -2\delta & \Omega_S \\ \Omega_P & \Omega_S & -2\Delta \end{bmatrix} $$
$$P= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} $$
$$Q= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$
Where I am trying to eliminate the highest energy state and get an effective two level system.
I expect to get the effective matrix seen in textbooks
$$H_{eff}= \begin{bmatrix} \frac{\Omega_P^2}{2\Delta} & \frac{\Omega_P \Omega_S}{2\Delta} \\ \frac{\Omega_P \Omega_S}{2\Delta} & \frac{\Omega_S^2}{2\Delta}-2\delta \end{bmatrix} $$
However, attempting to use the adiabatic transformation above seems to be impossible because $QHQ$ is singular. Clearly there's something mistaken here, but what is it?
Edit: Seems the equation does work provided that one takes the inverse as $$Q\frac{1}{QHQ}Q \rightarrow \lim_{\epsilon \to 0}Q\frac{1}{QHQ + \epsilon I}Q$$ Presumably this means that this is actually a limiting value of the Green function (resolvent).
