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

The gist The same cavity–atom platform can realize both emergent ladder spectra and hopping-induced localization in excitation-number space.

Model and Mapping

The system is modeled as a single microwave resonator coupled to an effective two-level artificial atom, analyzed beyond the rotating-wave approximation using a generalized quantum Rabi Hamiltonian (Page 2). The main conceptual step involves recasting the cavity-atom problem in the excitation-number basis, viewing it as motion on an effective one-dimensional lattice with parity-resolved sectors and photon-number-dependent hopping amplitudes (Page 1). This mapping leads to two effective Hamiltonians depending on the parity subspace:

1. The odd-parity subspace Hamiltonian is given by Ho = X N n=1 [λ1 √n¯ + 2n − 1(2n − 1iooh2n) + λ2 √n¯ + 2n(2niooh2n + 1)] + H.c. (Page 3).

2. The even-parity subspace Hamiltonian is He = X N n=1 [λ2 √n¯ + 2n − 1(2n − 1ieeh2n) + λ1 √n¯ + 2n(2nieeh2n + 1)] + H.c. (Page 3).

Large-Photon-Number Regime

In the large-photon-number regime, where n¯ >> 2N, the effective excitation-number lattice realizes a dimerized SSH-type structure subject to a uniform Wannier–Stark potential (Page 3). The effective Hamiltonian is decomposed into Hstark + H∆ + Homega, where:

1. The dominant term is the Wannier–Stark Hamiltonian Hstark = Hunif + Hω, which generates a uniform energy gradient leading to Wannier–Stark localization (Page 3).

2. The alternating hopping term H∆ introduces SSH-type dimerization (Page 3).

3. The staggered on-site potential Homega originates from the atomic degree of freedom and further couples the two sublattices (Page 3).

The spectrum organizes into one or two Wannier–Stark ladders depending on the effective perturbation parameter γ, which is defined by ratios among ω, omega, and coupling strengths (Page 4). The three representative cases are:

** γ = 0: the spectrum retains a single equally spaced ladder with level spacing ω (Page 4). This leads to undamped Bloch oscillations with a fundamental period TB = 2π/ω (Page 4). **

**/γ = ω/2: the spectrum exhibits extensive double degeneracy and an emergent ladder with spacing 2ω, resulting in frequency-doubled dynamics with a period T(2)B = π/ω (Page 4). **

**/γ ≠ 0, ω/2: the spectrum consists of two offset ladders with distinct spacings E1 and E2 satisfying E1 + E2 = 2ω, leading to Bloch–Zener oscillations characterized by a minimal revival time TBZ = 1/1/T1 − 1/T2 (Page 4). The dynamics can be expressed as Φ(t)i = cos(γt)ϕ0 (t)i − isin(γt)ϕ0(t + TB/2)i, where the relative weights are periodically exchanged at a rate controlled by the effective coupling γ (Page 4). **

Small-Photon-Number Regime and Localization

In the small-photon-number regime, the hopping amplitudes retain a strong photon-number dependence, causing the effective lattice to become strongly inhomogeneous (Page 6). The resulting hopping deformation reshapes the density of states and enhances collective localization (Page 6).

1. Hopping-deformed effective lattice: The Hamiltonian is written as Heff,2 = Hinc + Hω + Homega, where t1(n) and t2(n) increase monotonically with the cell index n (Page 6). This structure is described by a local SSH lattice whose parameters vary with position (Page 6).

2. Turning-point localization: The local turning-point criterion for propagation is given by χn(E) ≡ [[E − Vc(n)]2 − ∆z(n) − t22(n) − t12(n + 1) + 2t2(n)t1(n + 1) cos k] × [2t2(n)t2(n + 1)]−1 (Page 7). Localization occurs when χ n(E) = 1, which is not disorder induced but arises from a deterministic turning-point structure generated by the position dependence of the local SSH bands (Page 7).

3. Spectral reconstruction: The density of states D(E) is reshaped by hopping deformation, with Dloc(E, n) = X ν=± Z π −π dk/2π δ[E − Eν (n, k)] (Page 7). This reshaping anticipates substantial changes in eigenstate structure and transport (Page 7).

Dynamical Signatures

The real-time dynamics display direction-dependent anomalous diffusion together with Bloch-like revivals and beat phenomena in selected parameter windows (Page 1). In the small-photon regime, the left-right asymmetry of the diffusion is not a consequence of any asymmetry in the global spectrum but follows from the interplay between energy-resolved spatial bias and position dependence of local kinetic scale (Page 8). The mean-square displacement MSD shows that ballistic growth is driven almost entirely by the strong-hopping region, while sub-ballistic oscillatory behavior occurs on the weak-hopping side (Page 9). The final observation is that counterrotating coupling acts as a mechanism for engineering synthetic transport landscapes with experimentally accessible signatures (Page 11).

The paper concludes that the same microscopic platform supports two qualitatively distinct regimes controlled by the photon number, demonstrating how counterrotating coupling reorganizes excitation-number space and produces experimentally testable transport signatures (Page 10). The results establish a unified picture of how counter-rotating light-matter coupling reorganizes excitation-number space and produces experimentally testable transport signatures (Page 10). This identifies superconducting cavity–atom architectures as a promising platform for exploring emergent ladder physics, hopping-induced localization, and controlled nonequilibrium dynamics within a single light–matter setting (Page 11). The remainder of the paper is organised as follows: In Sec. II, we introduce the Hamiltonian for a superconducting resonator coupled to a two-level artificial atom and map it onto an effective tight-binding problem in the excitation-number basis (Page 2). Sec. III is devoted to the large-photon-number regime, in which we demonstrate that the effective excitation-number lattice realizes a dimerized SSHtype structure subject to a uniform Wannier–Stark potential (Page 3). In Sec. IV, we study the quantum dynamics of Heff,1 in the excitation-number lattice, where the interplay between Wannier-Stark localization and SSH-type dimerization enables flexible control of Bloch and BlochZener oscillations, quantitatively explaining the observed revival patterns (Page 4). Sec. V demonstrates that in the small-photon-number regime, the effective excitation-number lattice becomes strongly inhomogeneous, and the clean Bloch-type motion is replaced by state-dependent transport governed by the detailed structure of the spectrum, the density of states, and the spatial organization of the eigenstates (Page 6). The remainder of the paper is organised as follows: In Sec. II, we introduce the Hamiltonian for a superconducting resonator coupled to a two-level artificial atom and map it onto an effective tight-binding problem in the excitation-number basis (Page 2). Sec. III is devoted to the largephoton-number regime, in which we demonstrate that the effective excitation-number lattice realizes a dimerized SSHtype structure subject to a uniform Wannier–Stark potential (Page 3). In Sec. IV, we study the quantum dynamics of Heff,1 in the excitation-number lattice, where the interplay between Wannier-Stark localization and SSH-type dimerization enables flexible control of Bloch and BlochZener oscillations, quantitatively explaining the observed revival patterns (Page 4). Sec. V demonstrates that in the small-photon-number regime, the effective excitation-number lattice becomes strongly inhomogeneous, and the clean Bloch-type motion is replaced by state-dependent transport governed by the detailed structure of the spectrum, the density of states, and the spatial organization of the eigenstates (Page 6). We conclude in with a summary of our findings and a discussion of promising future directions (Page 2). The remainder of the paper is organised as follows: In Sec. II, we introduce the Hamiltonian for a superconducting resonator coupled to a two-level artificial atom and map it onto an effective tight-binding problem in the excitation-number basis (Page 2). Sec. III is devoted to the largephoton-number regime, in which we demonstrate that the effective excitation-number lattice realizes a dimerized SSHtype structure subject to a uniform Wannier–Stark potential (Page 3). In Sec. IV, we study the quantum dynamics of Heff,1 in the excitation-number lattice, where the interplay between Wannier-Stark localization and SSH-type dimerization enables flexible control of Bloch and BlochZener oscillations, quantitatively explaining the observed revival patterns (Page 4). Sec. V demonstrates that in the small-photon-number regime, the effective excitation-number lattice becomes strongly inhomogeneous, and the clean Bloch-type motion is replaced by state-dependent transport governed by the detailed structure of the spectrum, the density of states, and the spatial organization of the eigenstates (Page 6).

1. Small-photon-number regime dynamics analysis: The improved AI can predict direction-dependent anomalous diffusion together with Bloch-like revivals and beat phenomena in selected parameter windows by analyzing the hopping deformation which reshapes the density of states and generates an energy resolved spatial bias of the eigenstates.

2. Large-photon-number regime spectral prediction: The system can quantitatively explain the observed revival patterns by mapping the dynamics to a dimerized SSH-type chain subject to a linear energy gradient, specifically predicting whether the spectrum realizes one or two Wannier–Stark ladders.

3. Parameter space navigation for transport regimes: The AI can identify which dynamical regime is active (single-ladder regime, degenerate ladder and frequency-doubled oscillations, or two-ladder regime) based on tuning parameters like the effective coupling ratio γ, allowing it to predict the resulting Bloch oscillations (e.g., single-frequency Bloch oscillations vs. Bloch–Zener oscillations).

4. Experimental parameter optimization: Given target transport signatures (e.g., a specific beat pattern or localization), the system can suggest optimal control over parameters such as the ratio between coupling strengths and frequencies, leveraging the analysis of how the competition between the alternating hopping amplitudes and the staggered on-site potential controls these features.