Emergent SSH physics and localization in a cavity--atom system beyond the rotating-wave approximation

We investigate a cavity–atom system beyond the rotating-wave approximation and recast its dynamics as an effective tight-binding problem in the excitation-number basis. The same experimentally relevant light–matter platform exhibits two distinct regimes controlled by the photon number. In the large-photon-number regime, the hopping amplitudes become nearly uniform and the effective lattice reduces to a dimerized SSH-type chain subject to a linear energy gradient. The spectrum then organizes into one or two Wannier–Stark ladders, giving rise to controllable Bloch and Bloch–Zener oscillations and quantitatively explaining the observed revival patterns. In the small-photon-number regime, the intrinsic $\sqrt{\bar{n}+(\cdots )}$ dependence produces a pronounced hopping deformation, namely dimerized couplings that increase along the excitation-number lattice. This deformation reshapes the density of states, enhances collective localization, and generates an energy-resolved spatial bias of the eigenstates. The corresponding real-time dynamics displays direction-dependent anomalous diffusion together with Bloch-like revivals and beat phenomena in selected parameter windows. Our results show that the same cavity–atom platform can realize both emergent ladder spectra and hopping-induced localization in excitation-number space.