Eigenoperator approach to Schrieffer-Wolff perturbation theory and dispersive interactions

Modern quantum physics is very modular: we first understand basic building blocks (``XXZ Hamiltonian'' ``Jaynes-Cummings'' etc.) and then combine them to explore novel effects. A typical example is placing known systems inside an optical cavity. The Schrieffer-Wolff perturbation method is particularly suited for dealing with these problems, since it casts the perturbation expansion in terms of operator corrections to a Hamiltonian, which is more intuitive than energy level corrections, as in traditional time-independent perturbation theory. However, the method lacks a systematic approach.% and has largely remained a niche topic. In these notes we discuss how \emph{eigenoperator decompositions}, a concept largely used in open quantum systems, can be employed to construct an intuitive and systematic formulation of Schrieffer-Wolff perturbation theory. To illustrate this we revisit various papers in the literature, old and new, and show how they can instead be solved using eigenoperators. Particular emphasis is given to perturbations that couple two systems with very different transition frequencies (highly off-resonance), leading to the so-called dispersive interactions.