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Abstract

By applying the Born-Huang expansion, originally developed for coupled nucleus-electron systems, to the full nucleus-electron-photon Hamiltonian of non-relativistic quantum electrodynamics (QED) in the long-wavelength approximation, we deduce an exact set of coupled equations for electrons on photonic energy surfaces and the nuclei on the resulting polaritonic energy surfaces. Since the photonic surfaces and the corresponding non-adiabatic coupling elements can be solved analytically, the resulting expansion can be brought into a compact form which allows us to analyze certain aspects of coupled nucleus-electron-photon systems in a simple and intuitive manner. Furthermore, we discuss structural differences between the exact quantum treatment and Floquet theory, that along this line existing implementations can be adjusted to incorporate QED and how standard drawbacks of Floquet theory can be overcome. We then highlight, by assuming that the relevant photonic frequencies of a prototypical cavity QED experiment are in the energy range of the electrons, how from this generalized Born-Huang expansion an adopted Born-Oppenheimer approximation for nuclei on polaritonic surfaces can be deduced. By restricting the basis set of this generalized Born-Oppenheimer approximation we span the ark from quantum chemistry to quantum optics by recovering simple models of coupled matter-photon systems employed in quantum optics and polaritonic chemistry. We finally highlight numerically that simple few level models can lead to physically wrong predictions, even in the weak coupling regime, and show how the presented derivations from first principles help to check and derive physically grounded simplified models.