This is simply a photonic Hamiltonian added to a matter Hamiltonian with an effective mass scaling in the kinetic energy and a shift in coordinates in the potential energy. Note that the coupling is mediated by the shifting of each x j by the photonic operator weighted by an effective coupling term ξ g . However, |ξ g (g)| has a finite peak and then asymptotically approaches zero for large |g|. In other words, in this representation the photonic and matter degrees of freedom asymptotically decouple at arbitrarily high coupling strength. For this reason, this representation was put forward as a convenient Hamiltonian when considering systems in the deep strong coupling regime and beyond. This AD Hamiltonian has also been expanded upon in recent work for solid-state materials in reciprocal space in Ref. 53, but is beyond the scope of this review as we are focused on molecular QED.