What is the regime under which we may consider quantum optics description of light a good approximation of a more correct theory such as QED?
By quantum optics I mean describing the electromagnetic field as a collection of harmonic oscillators, and its interaction with particles through the Hamiltonian $$ H = \frac{(p-A)^2}{2m} + V_{coul} + H_{free} $$ where $H_{free}$ is a sum of harmonic oscillators.
The second-quantised description of the electromagnetic field in terms of oscillators holds in QED as well. The part that is modified is the single particle description of charged particles. In other words, (virtual and real) pair-creation is permitted in QED. So for energy scales less than $2mc^2$ as well as low intensities (see Schwinger limit), where pair-creation is not possible or suppressed, one can work with a fixed number of charged particles.
(An intermediate step to QED is to go from the non-relativistic Schrodinger equation to the Dirac (or Klein-Gordon) equation. The breakdown of this description is illustrated by the Klein paradox.)
(Edited to incorporate @curiousone's comment.)
According to wikipedia, Quantum optics is study of light and its interaction of matter in both semi-classical and quantum regime.
The semi-classical description is used when we are concerned with only average effects like intensity changes of light and population inversion in lasers. In this description light is treated as a classical electromagnetic wave and atom is treated as quantum mechanical object. All diffraction and interference phenomenon of light can be described by treating light as a classical electromagnetic wave.
But if we are concerned with short time fluctuations of light and higher order correlation functions of intensity we should use quantum description of light, i.e light is composed of discrete quanta called photons. These photons are created and annihilated by atoms and molecules. The creation and annihilation of photons is described by ladder operators ($a$ and $a^{\dagger}$ respectively) of a harmonic oscillator. This experiment will tell you more about quantum description of light.
Quantum Electrodynamics (QED) is a study of interactions between elementary particles at relativistic speeds (Very high energies) or to explain very tiny shifts in energy levels of atoms called Lamb shift. Here the interaction between the charged particles happens by exchanging photons. So the description of light here is same as the above-mentioned quantum description of light. These photons of QED are usually called virtual photons because they cannot be directly detected. But a real photon can also be produced when matter and anti-matter combine to give a photon. Also a very high-energy (high-frequency) photon called gamma ray photon can spontaneously create matter and anti-matter pair of particle in presence of a nucleus. This is the realm of QED.
