Ab initiononrelativistic quantum electrodynamics: Bridging quantum chemistry and quantum optics from weak to strong coupling

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. This theory describes seamlessly many-body interactions between nuclei, electrons and photons including the quantum fluctuation of the electromagnetic field and provides a proper first-principle framework to describe QED-chemistry phenomena, namely polaritonic and cavity chemistry effects. 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 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, show how existing implementations of Floquet theory can be adjusted to adhere to QED and highlight 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 adapted Born-Oppenheimer approximation for nuclei on polaritonic surfaces can be deduced. This form allows a direct application of first-principle methods of quantum chemistry such as coupled-cluster or configuration interaction approaches to QED chemistry. By restricting the basis set of this generalized Born-Oppenheimer approximation we furthermore bridge quantum chemistry and 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 weak-coupling regimes, and show how the presented derivations from first principles help to check and derive physically reliable simplified models. * Electronic address: christian.schaefer@mpsd.mpg.de †

Relevance of the Quadratic Diamagnetic and Self-Polarization Terms in Cavity Quantum Electrodynamics

“…Quantum optical considerations on the other hand focus typically on the description of the transversal fields and thus strongly simplify the electronic structure. The resulting quantum-optical models are designed to predict specific photonic observables and are consequentially limited in their predictability for the matter subsystem [14,15]. Recent interest in strong-light matter interaction is calling now for a consistent treatment of those historically as complementary perceived limits.…”

“…We can acknowledge this failure and focus on reasonable domains in which predictions remain in reasonably close agreement, e.g. focus on resonant interactions [15,72]. Alternatively, we adjust the PZW transformation to compensate the reduced space accordingly [80], or consider as much of the space as possible which leads us into the realm of first-principles cavity QED.…”

“…This qualitative change is also represented in spatial observables, i.e., the self-polarization term favors a reduced polarizability and thus focuses charge density in domains where charge is already present [14][15][16]. The bilinear coupling, which furthermore scales with the frequency, is typically weaker affecting the ground state, features the contrary tendency and their competition will determine the qualitative distribution of charges inside the cavity [15,109]. The resulting consequences can e.g.…”

“…If for example nuclear vibrations are approximated as phonon modes, the non-linear Debye-Waller-term ∝ ∇ k ∇ k Ĥ has to be added to the bilinear interaction [110,111]. This term originates from the quadratic elements in a Born-Huang expansion [15] with the very same physical effects as the quadratic components Here the first term on the right-hand side corresponds to the Coulomb interaction, the second term corresponds to the self-polarization contribution [85]. We can further approximately distinguish between situations where the wavefunctions of the different constituents overlap strongly and situations where there is no strong overlap.…”

Molecular polaritons for controlling chemistry with quantum optics

“…Another promising trend in the field is development of (semi) ab-initio methods for nonadiabatic molecular polariton dynamics, as first proposed by Feist et al [175]. This integrative approach has been further developed by several groups [173,174,176,178,181,183,184,199,200]. The main strength of this methodology is its compatibility with conventional electronic structure and molecular dynamics techniques.…”