An approach to ab initio cavity quantum electrodynamics (QED) based on the reduced density matrix (RDM) is developed. The expectation value of the Pauli-Fierz Hamiltonian is expressed in terms of one- and two-body electronic and photonic RDMs, and the elements of these RDMs are optimized directly in polynomial time by semidefinite programming techniques, without knowledge of the full wave function. QED generalizations of important ensemble $N$-representability conditions are derived and enforced in this procedure. The resulting approach is applied to the description of classic ground-state strong-electron-correlation problems, augmented by the presence of ultrastrong light-matter coupling. First, we assess cavity-induced changes to the singlet-triplet energy gap of the linear oligoacene series; for a heptacene molecule, this gap can change by as much as 1.9 $\text{kcal}/\text{mol}$ (or 15%) when the molecule is aligned along the cavity mode polarization axis. We also explore the metal-insulator transition in linear hydrogen chains and demonstrate that strong electron-photon interactions increase the insulating character of these systems under all coupling strengths considered.

• Received 4 April 2022
• Revised 13 July 2022
• Accepted 27 October 2022

DOI:https://doi.org/10.1103/PhysRevA.106.053710