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Strong light-matter coupling provides a promising path for the control ofquantum matter where the latter is routinely described from first-principles.However, combining the quantized nature of light with this ab initio tool setis challenging and merely developing, as the coupled light-matter Hilbert spaceis conceptually different and computational cost quickly becomes overwhelming.In this work, we provide a non-perturbative photon-free formulation of quantumelectrodynamics (QED) in the long-wavelength limit, which is formulated solelyon the matter Hilbert space and can serve as an accurate starting point forsuch ab initio methods. The present formulation is an extension of quantummechanics that recovers the exact results of QED for the zero- andinfinite-coupling limit, the infinite-frequency as well as the homogeneouslimit and we can constructively increase its accuracy. We show how thisformulation can be used to devise approximations for quantum-electrodynamicaldensity-functional theory (QEDFT), which in turn also allows to extend theansatz to the full minimal-coupling problem and to non-adiabatic situations.Finally, we provide a simple local-density-type functional that takes thestrong coupling to the transverse photon-degrees of freedom into account andincludes the correct frequency and polarization dependence. This is the firstQEDFT functional that accounts for the quantized nature of light whileremaining computationally simple enough to allow its application to a largerange of systems. All approximations allow the seamless application to periodicsystems.

Making ab initio QED functional(s): Non-perturbative and photon-free effective frameworks for strong light-matter coupling