In this work, we provide an overview of how well-established concepts in the fields of quantum chemistry and material sciences have to be adapted when the quantum nature of light becomes important in correlated matter-photon problems. We analyze model systems in optical cavities, where the matter-photon interaction is considered from the weak-to the strong-coupling limit and for individual photon modes as well as for the multimode case. We identify fundamental changes in Born-Oppenheimer surfaces, spectroscopic quantities, conical intersections, and efficiency for quantum control. We conclude by applying our recently developed quantum-electrodynamical density-functional theory to spontaneous emission and show how a straightforward approximation accurately describes the correlated electron-photon dynamics. This work paves the way to describe matter-photon interactions from first principles and addresses the emergence of new states of matter in chemistry and material science.QED chemistry | quantum electrodynamical density functional theory | adiabatic polariton surfaces | local control | optimized effective potential N ovel experimental possibilities have allowed scientists to obtain new insights into how photons interact with matter and how these interactions correlate photonic and particle degrees of freedom. Such experiments show, for example, an increase of the conductivity in organic semiconductors through hybridization with the vacuum field (3), strong shifts of the vibrational frequencies by the coupling of molecular resonators with a microcavity mode (4), nonclassical single photon-phonon correlations (5), the control of spin relaxations using an optical cavity (6), the enhancement of Raman scattering from vibropolariton states (7,8), changes of chemical reactivity (9, 10), single-molecule strong coupling (11), sampling of vacuum fluctuations (12), strong exciton-photon coupling of light-harvesting complexes (13), strong long-range atom-atom interactions mediated by photons (14), attractive photonic states (15, 16), or superradiance for atoms in photonic crystals (17). All these results indicate the appearance of new states of matter and subsequently a change in the chemical properties of the matter system (18-21), if the quantum nature of light becomes important. For example, in so-called strong-coupling situations, which are nowadays of central interest in the fields of circuit quantum electrodynamics (circuit QED) (22-24) or cavity QED (25,26). Whereas the analyses of such experiments are routinely performed with the help of simplified (few-level) models that are able to capture the essential physics, for the (quantitative) prediction of properties of complex multiparticle systems coupled to photons, methods that can treat such coupled boson-fermion situations from first principles seem worthwhile (1, 2, 27-31). On the other hand, the strong coupling to photons can challenge our conventional understanding of electronic structures and allows us to study the influence of the quantum nature of light on ch...

In this work, we give a comprehensive derivation of an exact and numerically feasible method to perform ab initio calculations of quantum particles interacting with a quantized electromagnetic field. We present a hierarchy of density-functional-type theories that describe the interaction of charged particles with photons and introduce the appropriate Kohn-Sham schemes. We show how the evolution of a system described by quantum electrodynamics in Coulomb gauge is uniquely determined by its initial state and two reduced quantities. These two fundamental observables, the polarization of the Dirac field and the vector potential of the photon field, can be calculated by solving two coupled, nonlinear evolution equations without the need to explicitly determine the (numerically infeasible) many-body wave function of the coupled quantum system. To find reliable approximations to the implicit functionals, we present the appropriate Kohn-Sham construction. In the nonrelativistic limit, this density-functional-type theory of quantum electrodynamics reduces to the densityfunctional reformulation of the Pauli-Fierz Hamiltonian, which is based on the current density of the electrons and the vector potential of the photon field. By making further approximations, e.g., restricting the allowed modes of the photon field, we derive further density-functional-type theories of coupled matter-photon systems for the corresponding approximate Hamiltonians. In the limit of only two sites and one mode we deduce the appropriate effective theory for the two-site Hubbard model coupled to one photonic mode. This model system is used to illustrate the basic ideas of a density-functional reformulation in great detail and we present the exact Kohn-Sham potentials for our coupled matter-photon model system.

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