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

C. Schäfer, Florian Buchholz+3 more|arXiv (Cornell University)|Jun 14, 2021

Strong Light-Matter InteractionsPhysics and Astronomy88 references71 citations

TL;DR

This paper introduces a non-perturbative, photon-free formulation of ab initio quantum electrodynamics (QED) that operates solely on the matter Hilbert space, enabling accurate and computationally feasible simulations of strong light-matter coupling. The framework recovers exact QED results in key limits—zero and infinite coupling, infinite frequency, and homogeneous systems—while enabling the construction of a simple, frequency- and polarization-dependent local-density-type functional for quantum-electrodynamical density-functional theory (QEDFT).

ABSTRACT

Strong light-matter coupling provides a promising path for the control of quantum matter where the latter is routinely described from first-principles. However, combining the quantized nature of light with this ab initio tool set is challenging and merely developing, as the coupled light-matter Hilbert space is conceptually different and computational cost quickly becomes overwhelming. In this work, we provide a non-perturbative photon-free formulation of quantum electrodynamics (QED) in the long-wavelength limit, which is formulated solely on the matter Hilbert space and can serve as an accurate starting point for such ab initio methods. The present formulation is an extension of quantum mechanics that recovers the exact results of QED for the zero- and infinite-coupling limit, the infinite-frequency as well as the homogeneous limit and we can constructively increase its accuracy. We show how this formulation can be used to devise approximations for quantum-electrodynamical density-functional theory (QEDFT), which in turn also allows to extend the ansatz to the full minimal-coupling problem and to non-adiabatic situations. Finally, we provide a simple local-density-type functional that takes the strong coupling to the transverse photon-degrees of freedom into account and includes the correct frequency and polarization dependence. This is the first QEDFT functional that accounts for the quantized nature of light while remaining computationally simple enough to allow its application to a large range of systems. All approximations allow the seamless application to periodic systems.

Motivation & Objective

• To develop a non-perturbative, photon-free formulation of ab initio QED that avoids the exponential scaling of coupled matter-photon Hilbert spaces.
• To construct a matter-only theory that recovers exact QED results in fundamental physical limits such as zero and infinite coupling.
• To enable the derivation of accurate, computationally tractable approximations for quantum-electrodynamical density-functional theory (QEDFT).
• To provide a systematically improvable framework that accounts for quantized light effects while remaining applicable to periodic systems and non-adiabatic dynamics.

Proposed method

• Proposes a photon-free QED reformulation by expressing photonic operators in terms of matter degrees of freedom, embedding dominant field fluctuations into the matter sector.
• Uses a truncated basis expansion of the full QED problem to systematically improve accuracy and converge toward exact QED solutions.
• Derives an effective matter-only Hamiltonian that retains non-perturbative light-matter correlations and recovers exact results in limiting cases.
• Constructs an orbital-dependent approximation for QEDFT based on the same matter-only ansatz, ensuring consistency with the non-perturbative framework.
• Introduces a simple local-density-type functional that includes frequency and polarization dependence of the photonic field via matter-based parameters.
• Ensures compatibility with periodic systems and extends to non-adiabatic and minimal-coupling scenarios through the effective matter-only formulation.

Experimental results

Research questions

• RQ1 Can a photon-free, non-perturbative QED framework be constructed that accurately captures strong light-matter coupling without explicit photonic degrees of freedom?
• RQ2 Does the proposed matter-only formulation recover exact QED results in key physical limits such as zero coupling, infinite coupling, and the homogeneous electron gas?
• RQ3 Can this framework be used to derive a practical, computationally efficient functional for quantum-electrodynamical density-functional theory (QEDFT)?
• RQ4 How can the accuracy of the effective theory be systematically improved while preserving physical consistency?
• RQ5 Can the framework be extended to non-adiabatic and minimal-coupling problems in periodic systems?

Key findings

• The proposed photon-free QED formulation exactly recovers the QED solution in the zero- and infinite-coupling limits, as well as in the infinite-frequency and homogeneous limits.
• The framework is systematically improvable via a basis expansion of the full QED problem, converging to exact results with increasing truncation.
• A local-density-type functional for QEDFT is constructed that includes frequency and polarization dependence of the photonic field through matter-based parameters.
• The effective theory enables seamless application to periodic systems and extends to non-adiabatic and minimal-coupling scenarios.
• The approach provides a computationally feasible, ab initio starting point for strong light-matter coupling that avoids the exponential scaling of the coupled matter-photon Hilbert space.
• The method offers a consistent, non-perturbative alternative to standard quantum mechanics, unifying light and matter in a single effective matter-only description.