  We develop a new 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 new Hamiltonian is obtained by
applying a unitary gauge transformation on the p$\cdot$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 new 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 p$\cdot$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.
