Hierarchy of approximations for describing quantum light from high-harmonic generation: A Fermi-Hubbard model study

The quantum optical description of high-order harmonic generation where both the electrons of the generating medium and the driving and generated light fields are described quantum mechanically has been of significant interest in the past years. The quantum optical formulation leads to equations of motion for the generated light field in which the quantum optical field couples to the time-dependent current of the electronic medium irrespectively of the specifics of the electronic system being an atom, molecule, or solid. These equations of motion are not solvable for any realistic system and accurate and verified approximations are hence needed. In this work, we present a hierarchy of approximations for the equations of motion for the photonic state. At each level in this hierarchy, we compare it to the previous level justifying the validity using the Fermi-Hubbard model as an example of an electronic system with correlations. This model allows us to perform an accurate simulation of the electron motion of all the required states. We find that for the typical experimental situation of weak quantized-light-matter-coupling constant and at intensities well below the damage threshold, an explicit expression for the generated quantum light, referred to as the Markov-state approximation (MSA), captures the high-harmonic spectrum quantitatively and describes the single-mode quantum properties of the generated light as characterized by the Mandel-Q parameter and the degree of squeezing qualitatively.