  Most quantum theorists are familiar with different ways of describing the
effective quantum dynamics of a system coupled to external degrees of freedom,
such as the Born-Markov master equation or the adiabatic elimination.
Understanding the deep connection between these apparently unrelated methods
can be a powerful tool, allowing us to derive effective dynamics in
unconventional systems or regimes. This tutorial aims at providing quantum
theorists across multiple fields (e.g. quantum and atom optics, optomechanics,
or hybrid quantum systems) with a self-contained practical toolbox to derive
effective quantum dynamics, applicable to systems ranging from N-level emitters
to mechanical resonators. First, we summarize the projector approach to open
quantum systems and the derivation of the fundamental Nakajima-Zwanzig
equation. Then, we show how three common effective equations, namely the
Born-Markov Master Equation, the adiabatic elimination used in atom physics,
and a different adiabatic elimination used in sideband cooling, can be derived
from different perturbative expansions of the Nakajima-Zwanzig equation. We
also solve in detail two specific examples using this formalism, namely the
adiabatic elimination in a Lambda system and the effective equations of a
mechanical resonator cooled by an optical cavity.
