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By applying the Born-Huang expansion, originally developed for couplednucleus-electron systems, to the full nucleus-electron-photon Hamiltonian ofnon-relativistic quantum electrodynamics (QED) in the long-wavelengthapproximation, we deduce an exact set of coupled equations for electrons onphotonic energy surfaces and the nuclei on the resulting polaritonic energysurfaces. This theory describes seamlessly many-body interactions betweennuclei, electrons and photons including the quantum fluctuation of theelectromagnetic field and provides a proper first-principle framework todescribe QED-chemistry phenomena. Since the photonic surfaces and thecorresponding non-adiabatic coupling elements can be solved analytically, theresulting expansion can be brought into a compact form which allows us toanalyze aspects of coupled nucleus-electron-photon systems in a simple andintuitive manner. Furthermore, we discuss structural differences between theexact quantum treatment and Floquet theory and show how existingimplementations of Floquet theory can be adjusted to adhere to QED. From thisgeneralized Born-Huang expansion an adapted Born-Oppenheimer approximation fornuclei on polaritonic surfaces can be deduced. This form allows a directapplication of first-principle methods of quantum chemistry such ascoupled-cluster or configuration interaction approaches to QED chemistry. Byrestricting the basis set of this generalized Born-Oppenheimer approximation wefurthermore bridge quantum chemistry and quantum optics by recovering simplemodels of coupled matter-photon systems employed in quantum optics andpolaritonic chemistry. We finally highlight numerically that simple few-levelmodels can lead to physically wrong predictions, even in weak-coupling regimes,and show how the presented derivations from first principles help to check andderive physically reliable simplified models.

Ab-initio non-relativistic quantum electrodynamics: Bridging quantum chemistry and quantum optics from weak to strong coupling