We develop a theoretical framework for describing light-matter interactions in cavity quantum electrodynamics (QED), optimized for efficient convergence at arbitrarily strong coupling strengths and is naturally applicable to low-dimensional materials. This Hamiltonian is obtained by applying a unitary gauge transformation on the $\mathrm{p}·\mathrm{A}$ Hamiltonian, with a shift on both the matter coordinate and the photonic coordinate, then performing a phase rotation and transforming in the reciprocal space of the matter. By formulating the light-matter interaction in terms of an upper-bounded effective coupling parameter, this method allows one to easily converge eigenspectra calculations for any coupling strength, even far into the ultra-strong and deep-strong coupling regimes. We refer to this approach as the reciprocal asymptotically decoupled (RAD) Hamiltonian. The RAD Hamiltonian allows for a fast convergence of the polariton eigenspectrum with a much smaller matter and photon basis, compared to the commonly used $\mathrm{p}·\mathrm{A}$ or dipole gauge Hamiltonians. The RAD Hamiltonian also allows one to go beyond the commonly used long-wavelength approximation and accurately describes the spatial variations of the field inside the cavity, which ensures the conservation of momentum between light and matter.

• Received 14 November 2023
• Accepted 21 February 2024

DOI:https://doi.org/10.1103/PhysRevB.109.104305