Because there is no A(t) 2 term in this Hamiltonian, this Hamiltonian is tridiagonal in the photonic DOF. This significantly decreases the number of n blocks needed to converge eigenspectrum of this Hamiltonian, making this form more popular for modelling laser-driven systems. A detailed matrix form of the Floquet Hamiltonian can be found in Ref. 84, Chapter 9 (see Fig. .5). Fundamentally, Floquet theory depends on the assumption that the light field is strong enough that the photon number is reaching the infinite limit. In that limit, the matrix structure of Fig. (a) converges to that of Fig. (b) upon rescaling A 0 and subtracting a zero point energy. As discussed in Ref. 85, the Floquet picture in this limit can be thought of as the QED Hamiltonian when considering a range of Fock states that are sufficiently highly excited such that √ n ≈ √ n + δn, where δn is the number of Floquet blocks considered. Thus, under this limit of an intense field interacting with the matter, Floquet theory is a natural choice for strongly driven systems since numerical convergence can be reached for a few Floquet blocks. However, in the few photon limit, Floquet theory does break down, and the explicit QED treatment is necessary. It also suggests that the fundamental difference between the Floquet picture and the QED picture arises in the few photon limits. For example, when the hybrid system quickly explores several photon-dressed states |g, n + 1⟩ → |e, n⟩ through light-matter coupling and |e, n⟩ → |g, n⟩ through nonadiabatic coupling ⟨e|∇|g⟩, the system will explore the different photon number blocks, and the light-matter coupling strength scales with √ n due to the operator nature of the coupling term (â † + â). The Floquet picture, on the other hand, always exhibits the same light-matter coupling strength. This difference between the QED and Floquet pictures has been observed in a recent theoretical study .