Flat band mediated photon-photon interactions in 2D waveguide QED networks

TL;DR

This work shows that a Lieb-like lattice of nonlinear quantum emitters coupled to a 2D waveguide network hosts an energetically isolated flat band ($ε^{(1)}=0$) separated from two dispersive bands by a finite gap. The flat band arises from plane-wave–mediated long-range couplings and supports compact localized states that remain dark in finite arrays, even with nonlocal interactions. In the two-excitation sector, emitter nonlinearity induces effective photon–photon interactions: in the softcore regime, bound photon pairs form dispersive bound states within the gap and enable interaction-driven transport of localized excitations; in the hardcore limit, metastable exciton-like dressed states emerge from hybridization between flat and dispersive bands, with transport governed by the gap and edge losses. Collectively, the results illuminate how geometry, long-range couplings, and nonlinearity cooperate to realize highly correlated photonic states in flat-band systems, with potential realizations in optical or microwave waveguide networks and implications for flat-band superfluidity and topological photonics.

Abstract

We investigate a Lieb lattice of quantum emitters coupled to a two-dimensional waveguide network and demonstrate that this system supports an energetically isolated flat band, enabling localization despite the presence of long-range photon-mediated couplings. We then explore the two-excitation dynamics in both the softcore and hardcore interaction regimes, which arise from the nonlinearity of the emitters. In the softcore regime, we observe interaction-induced photon transport within the flat band, mediated by the formation of bound photon pairs. In the hardcore regime, corresponding to the two-level atom limit, we instead find the emergence of metastable exciton-like dressed states involving both flat and dispersive bands. Our findings highlight how the interplay between the collective behavior of emitters and effective photon-photon interactions can provide a platform for studying highly correlated photonic states in flat-band systems.

Figures (5)

• Figure 1: (a) A Lieb lattice of $N$ quantum emitters with lattice constant $d$ and intracell distance $a$ is coupled to a square grid of one-dimensional waveguides. The unit cell of the lattice consists of a corner site and two edge sites coupled to two waveguides and an individual waveguide, respectively. The quantum emitters are modeled as nonlinear resonators with on-site repulsive interaction $U$. (b) The single-excitation dispersion within the first Brillouin zone: two dispersion branches are separated by an isolated flat band at zero energy. Here, we set $k_0d = \pi$ and $a = \frac{d}{2}$.
• Figure 2: The components of the Quantum Geometric Tensor, as computed in Eq. \ref{['eq: Calculating geometric tensor']}, are shown as functions of the wavevector components $k_x$ and $k_y$. The first line corresponds to the result obtained for the Lieb lattice coupled to a two-dimensional waveguide network, with $k_0d=\pi$ and $a=d/2$, as discussed in the main text. To provide a direct comparison, we plotted in the second line the result for the nearest-neighbor where the two intracell are set to $t_1=1.2$ (coupling between A and B sites) and $t_2=0.8$ (coupling between B and C sites)
• Figure 3: The two-excitation dispersion in the softcore interactions regime within the isolated-band approximation calculated by diagonalizing the matrix given in Eq. \ref{['eq: interaction matrix']} as a function of the center-of-mass momentum $\mathbf{K}$. Panel (a) displays half of the full 2D Brillouin zone ($k_y<0$) for better resolution, while panel (b) shows the dispersion along the symmetry lines of the Brillouin zone. (c) Relative coordinate population distribution of the bound states having the interaction-induced dispersive bands shown in panls (a)-(b) at the $M,\Gamma$ and $X$ symmetry-points. The upper (lower) panel displays the upper (lower) branch of the bound states. The color map represents the probability distribution of finding the second excitation on different atoms in the lattice, given that the first excitation is located on the atom marked with the orange circle. The results were obtained for a system of $30\times 30$ unit cells.
• Figure 4: Time evolution of the state defined in Eq. \ref{['eq: FB initial state']} in the softcore interactions regime with $U = 0.1 \gamma$. (a) Initial state fidelity $\mathcal{F}_0(t)$ (blue solid line), flat-band subspace projection $\mathcal{P}_{FB}$ (dashed red line) and norm of the evolved state $\mathcal{N}(t)$ (black dotted line) as a function of time. (b) Snapshots of the excitation population evolution at times $t=0$ (left panel), $t=t_{min}\left[\mathcal{F}_0\right]$ (center panel) and $\gamma t=10^4$ (right panel), for two initially excited neighboring CLSs. The simulation was performed for a system size of $8\times 8$ unit cells. (c) Projection of the two-excitation state onto the interaction-induced dispersive bound states (red solid line) and onto the interacting non-dispersive states (gray dashed line) shown in Fig. \ref{['fig: two-excitation FB spectrum']}.
• Figure 5: (a) Same as in Fig. \ref{['fig: FB excitation population in weak interaction limit']}(a) in the hardcore interactions regime ($U\rightarrow\infty)$, for the initial condition defined in Eq. \ref{['eq: initial condition FB hardcore bosons']}. (b) Scaling of the fidelity oscillation frequency, $\omega_0$ for the initial condition defined in Eq. \ref{['eq: initial condition FB hardcore bosons']}, as a function of interaction strength $U$. The continuous line indicates the fit of the numerical data in the linear regime.