Multimode equilibrium approximations in light-matter systems from weak to strong coupling

TL;DR

This work develops and benchmarks three approximation strategies for equilibrium properties of matter strongly coupled to a multimode photonic environment within non-relativistic QED in the long-wavelength limit. By comparing to exact NRQED results for 1D atomic and molecular models and a 2D GaAs quantum ring, the authors show that an averaged-mode approach, NRQED$_{\text{ave}}$, most accurately reproduces ground-state energies and densities while greatly reducing computational cost. The dipole self-energy (M$+$DSE) and the lowest-mode (NRQED$_{\text{low}}$) schemes can fail to capture essential light-matter correlations or higher-frequency contributions, especially as the number of photon modes grows. The averaging strategy thus offers a practical, first-principles-compatible path to simulate complex quantum materials in realistic cavities, with broad implications for polaritonic chemistry and cavity-QED materials design, including potential extensions to excited-state phenomena.

Abstract

In this work, we detail different approaches to treat multi-mode photonic environments within non-relativistic quantum electrodynamics in the long-wavelength approximation efficiently. Specifically we show that for equilibrium properties of coupled light-matter systems, we can approximately capture the effects of multi-mode photonic environments on matter systems by either only keeping the polarization part of the electric field in the length-gauge formulation or by a few effective modes. We present a comprehensive set of approximation methods designed to accurately capture equilibrium phenomena in quantum light-matter systems across a range of complex photonic environments, from weak to strong coupling. These methods are applied to atomic and molecular models as well as to a two-dimensional quantum ring, demonstrating the versatility of our approach and laying the groundwork for first-principles simulations of real materials in cavity quantum electrodynamics.

Figures (9)

• Figure 1: The photon occupation of the combined ground-state for the atomic system coupled to a discretized continuum sampled with $N_{p}=250$ photon modes. The lowest modes couple strongest and the occupation increases with the coupling strength. The inset shows the same exponential decrease for higher-lying modes.
• Figure 2: Comparison of the correlated atomic ground-state energy for increasing number of modes showing a qualitative agreement between the result of NRQED and "NRQED$_\text{ave}$" while (M$+$DSE) deviates strongly for higher photon modes.
• Figure 3: The integrated atomic ground-state electron density difference between the NRQED result and the different approximations strategies, namely NRQED$_{\rm ave}$ (blue), NRQED$_{\rm low}$ (red), and $\rm (M+DSE)$ (green). We observe that $\rm (M+DSE)$ strongly deviates from the exact result for $N_p>50$, while NRQED$_{\rm low}$ has better performance but still performs worst than NRQED$_\text{ave}$. Thus, NRQED$_\text{ave}$ has the best performance when compared to the other methods.
• Figure 4: The integrated atomic ground-state electron density difference for different light-matter coupling strengths for (a) "NRQED$-$NRQED$_\text{ave}$", (b) "NRQED$-$NRQED$_\text{low}$" and (c) "NRQED$-$(M$+$DSE)" where the density difference increases with $\lambda_{\alpha}$. We observe that NRQED$_\text{ave}$ performs the best as the deviations from the exact NRQED are an order of magnitude smaller ($\sim 10^{-6}$) in comparison to the other methods ($\sim 10^{-5}$).
• Figure 5: Comparison of the dissociation energy of the ground-state of molecular H$_{2}$ for increasing number of photon modes where the (M$+$DSE) approximation strategy deviates from the exact NRQED result and the "NRQED$_\text{ave}$" qualitatively approximates the NRQED results.