Hybrid light-matter boundaries of graphene in a chiral cavity

Abstract

Recent advances in chiral cavities that can couple coherently to two-dimensional materials have opened a powerful route to reshape electronic topology without an external drive. Here we establish the bulk-boundary correspondence for graphene embedded in a circularly polarized cavity. By combining exact diagonalization (ED) of zigzag ribbons, a semi-analytic T-matrix for half-infinite lattices, and analytical insights from a Dirac-Jaynes-Cummings model, we show that (i) every light-matter interaction-induced gap hosts pairs of unidirectional light-matter edge currents depending on the Chern number of the band while some of them are even bright; (ii) these chiral states persist throughout the entire photon ladder; and (iii) their dispersion, localization length and photon distribution exhibit a universal scaling controlled by the light-matter interaction. Time-evolution simulations further demonstrate that a dark electronic edge excitation can be converted into a bright and unidirectionally propagating current that remains coherent over long time scales. Our results predict an experimental signature of the hybrid band topology and a blueprint for tunable chiral channels in next generation quantum optical solid-state devices.

Figures (3)

• Figure 1: Graphene strip in a chiral cavity with its bright and topological edges. (a) Conceptual set‐up: a graphene ribbon sits at the electric‐field antinode of a right‐handed cavity. The circularly polarized vacuum field breaks time‐reversal symmetry, imprinting a unidirectional edge channel (red glow) along each ribbon boundary. (b) ED spectrum for a zigzag ribbon of $N_y=1600$ sites, cavity frequency $\omega_c = 0.10t$, and coupling $g = 0.025t$ (Hilbert space truncated to two photons for convenience). The color scale shows the photon number expectation value of the state $\langle a^\dagger a\rangle$. Hybridization with the $n=0,1,2$ photon replicas opens interaction gaps; every gap hosts at least two chiral edge modes. Those involving a net photon exchange become strongly entangled (“bright”) and thus carry a finite number of photons. (c–d) Edge dispersions from the semi‑infinite T‑matrix formalism, zoomed into the two‑photon (c, upper) and one‑photon (d, lower) exchange gaps. Colored lines overlay the finite‑ribbon data from (b), demonstrating that the analytic T‑matrix fully captures the same bulk–boundary correspondence. For the details of the latter, see text. (e) Real‐space edge‐state profile at $k_x=2.4$: the transparency of the surface plot gives $|\psi(x,y)|^2$, while the color scale maps the local photon occupation. At the boundary, the state is in an equal superposition of $n=0$ and $n=1$ photons, yielding $\langle n\rangle\approx0.5$.
• Figure 2: Coherent formation of a bright chiral edge mode from a dark edge state and build-up of light–matter entanglement. Time evolution is launched from a dark edge state, vacuum Fock state, at fixed momentum $k_x = 2.4$ (same system parameters as in Fig. \ref{['fig:fig1']}). (a,b) Photon number distribution $p_n(t)$ in relatively short and long times. The initially product state quickly spreads over the neighboring Fock sectors as the bright chiral edge mode is populated. Because the edge mode is in a superposition of very few exact topological edge modes of the full Hamiltonian, the ensuing Rabi-like oscillations remain phase-coherent over the entire simulation window. (c,d) von Neumann entropy $S(t)$ of the reduced photonic density matrix in relatively short and long times. $S(t)$ oscillates around the singlet value $S_{\mathrm{singlet}}=\ln 2$ (red dashed line), signaling a maximally entangled cat state of one photon shared between light and matter. Note that the same qualitative behavior is obtained at any momentum that hosts a hybrid edge mode; if no such mode exists, $S(t)$ would remain zero. The orange line is the long-time average of $S(t)$ in (d).
• Figure 3: Localization length and the velocity of a bright edge mode. (a) For $\omega_c=0.1 t$, we compare $k_{x,0}$, the momentum of the crossing between left and right localized edge mode, calculated using both the TM method for a semi-infinite graphene ribbon and the continuum DJC model. Similarly, in (b) the localization length $\xi$ at $k_{x,0}$ and in (c) the edge mode velocity determined by these two methods. The localization length decreases with increasing light-matter interaction strength $g$, while the magnitude of the velocity increases. Agreement between analytical continuum model and the exact numerical results is perfect up to ultrastrong coupling regime. (d) The bright edge modes are exponentially localized at any $g$.