Repulsive Inverse-Distance Interatomic Interaction from Many-Body Quantum Electrodynamics

TL;DR

The paper introduces a many-body QED (MB-QED) framework that combines the quantum Drude oscillator (QDO) model with the many-body dispersion (MBD) approach to describe collective electronic fluctuations in coupled atoms. By diagonalizing the matter Hamiltonian into molecular-plasmon normal modes and incorporating their coupling to the quantized electromagnetic field, the authors derive a radiative interatomic interaction that, in the near zone, contains a persistent repulsive term scaling as $\sim 1/R$. This MB-QED $1/R$ interaction, proportional to $\alpha_{\mathrm{fsc}}^{3}$, arises from the coupling between molecular plasmons and virtual photons and can surpass gravitational forces in microscopic-scale quantum tests, with measurable impact in precision Casimir and short-range gravity experiments. The Argon dimer serves as a concrete example, showing that the $1/R$ term can contribute up to about 10% of the dispersion energy in the intermediate regime, and they provide both analytic and numerical evidence for the effect, along with a discussion of its interpretation as an emergent, source-like parameter. The work also discusses limitations (perturbative, model-dependent renormalization) and outlines non-perturbative avenues for future refinement to connect with renormalization concepts in QED.

Abstract

Interactions between objects can be classified as fundamental or emergent. Fundamental interactions are either extremely short-range or decay inversely with the separation distance, such as the Coulomb potential between charges or the gravitational attraction between masses. In contrast, emergent quantum van der Waals (vdW) and Casimir interactions decay considerably faster ($R^{-6}$ or $R^{-7}$) with distance $R$. Here we apply perturbative quantum electrodynamics (QED) to a many-body (MB) system of atoms modeled as charged harmonic oscillators, and reveal a persistent inverse-distance MB-QED interaction stemming from the coupling between virtual photons and molecular plasmons in the non-retarded regime. This interaction, scaling with the third power of the fine-structure constant, is reminiscent of the Lamb shift for a single atom. Although weaker than vdW forces, this MB-QED $R^{-1}$ interaction may substantially surpass gravitational attraction in future experiments probing quantum gravity at microscopic scales.

Figures (2)

• Figure 1: a) Two quantum Drude oscillators (QDOs) interacting with one another via dipole-dipole coupling $\widehat{V}_{\mathrm{DD}}$, and with the electromagnetic field (EMF) through the minimal coupling term $\widehat{\bm{p}}_{a} \cdot \widehat{\bm{A}}(\bm{R}_a)$ that results in the exchange of transverse photons $\gamma_{\perp}$. b) QDOs normal modes resulting from diagonalization of the many-body dispersion (MBD) Hamiltonian, i.e.$\widehat{H}_{\mathrm{MBD}}=\widehat{H}_{\rm QDO}^{(1)}+\widehat{H}_{\rm QDO}^{(2)}+\widehat{V}_{\mathrm{DD}}$ (see SM for the diagonalization). There are six normal modes, including two longitudinal modes ($\pi_\pm^z$) and four transverse modes ($\pi_\pm^x$, and two other modes with $\pi_\pm^y$ that are not shown here). These modes interact with the EMF, exchanging transverse photons $\gamma_{\perp}$, via $\widehat{H}_{\text{int}}=-\sum_{L,i}\widehat{\pi}_{L,i} \mathcal{A}_{L,i}$ as in Eq. \ref{['def:interaction_term']}. c) Interaction energy $\Delta V^{\text{int}}_{12}$ defined in Eq. \ref{['eq:DeltaV_int']} as a function of the inter-QDO distance $R$ for different values of the cutoff ratio $k_M/\lambda_{\rm QDOM}^{-1}$, where $\lambda_{\rm QDOM} = \max_a \sqrt{\hbar/2m\omega_a}$ (see main text). The violet triangles, red squares, green diamonds, orange triangles, and blue circles correspond respectively to $k_M/\lambda^{-1}_{\rm QDOM}=2 \cdot 10^{-2},\ 10^{-2},\ 5 \cdot 10^{-3},\ 2 \cdot 10^{-3}$, and $10^{-3}$. The dashed black line shows the fit $y(R) = 8.62 \times 10^{-8} R^{-1.04}$, while the solid black line corresponds to $y(R) = 4.29 \times 10^{-15} R^{-1.08}$. Notice that the deviations from the $1/R$ behavior at short range and low frequencies arise from the higher-order terms, particularly the $R^{-9}$ term in \ref{['def:R_behavior']}, which dominates over $1/R$ for smaller values of the cut-off wavenumber $k_M$ (see Sec. V in the SM for a detailed analysis). d) Contour plot of $\log_{10}[\Delta V(c_1 R^{-1}) / c_6 R^{-6}]$ as a function of QDO frequencies and interatomic distance $R$. This illustrates the crossover between the many-body QED correction and the van der Waals interaction across spatial and frequency scales.
• Figure S1: Error analysis in the approximation of $\mathcal{I}_{\rm appr}(\bar{k};\gamma,\tau,\eta)$.