Compact cavity-dressed Hamiltonian framework at arbitrarily strong light-matter coupling

We present a nonperturbative Hamiltonian mapping method for quantum systems strongly coupled to a quantized field mode (cavity), yielding compact closed-form representations of hybrid light-matter systems, allowing efficient studies of equilibrium and dynamical properties. The mapping builds on an entangling transformation of photonic and atomic degrees of freedom. By truncating the resulting cavity-dressed Hamiltonian (CDH) to successively larger excitation sectors, we construct a series of compact models that converge to the exact limit, outperforming conventional approaches even in the challenging resonant and ultrastrong light-matter regime. We benchmark the CDH framework on the quantum Rabi model, obtaining accurate spectra and thermal observables across weak and strong coupling regimes. We further apply the method to the Rabi model, highlighting its computational advantages for nonunitary dynamics. As a nontrivial application, we analyze the Dicke-Heisenberg lattice model, determine its phase diagram under resonant and strong light-matter coupling, and reveal cavity-mediated spin correlations analytically and numerically, achieving substantial computational savings over brute-force simulations. The closed form and compactness of the CDH provide both physical insight and improved computational efficiency in studies of strongly coupled hybrid light-matter systems.