Strongly Interacting Photons in 2D Waveguide QED

One-dimensional confinement in waveguide quantum electrodynamics (QED) plays a crucial role to enhance light-matter interactions and to induce a strong quantum nonlinear optical response. In two or higher-dimensional settings, this response is reduced since photons can be emitted within a larger phase space, opening the question whether strong photon-photon interaction can be still achieved. In this study, we positively answer this question for the case of a 2D square array of atoms coupled to the light confined into a two-dimensional waveguide. More specifically, we demonstrate the occurrence of long-lived two-photon repulsive and bound states with genuine 2D features. Furthermore, we observe signatures of these effects also in free-space atomic arrays in the form of weakly subradiant in-band scattering resonances. Our findings provide a paradigmatic signature of the presence of strong photon-photon interactions in 2D waveguide QED.

Figure 1

(a) A square array of two level atoms is coupled to the light confined into a 2D photonic waveguide. (b) Single photon dispersion relation for $k_{0}d=(\pi /2)$. Upper right panel: first Brillouin zone and $\mathrm{\Gamma }$, $X$, $M$ symmetry points. Lower right panel: Band gap energy as a function of the interatomic distance $k_{0}d$. The gap closes at $k_{0}d>\pi$. The dashed orange lines indicate the dispersion’s divergences at $|\mathbf{k}|=k_0$.

Figure 2

(a) Two-excitation spectrum as a function of the center-of-mass momentum for $k_{0}d=(\pi /2)$. Blue dots represent unbound states while the red lines and the orange dot represent the bound states. The dashed black boxes indicate the regions where the repulsive states lie. (b) Relative coordinate population distribution, $p(r_x,r_y)$, of two kinds of repulsive states at the $\mathrm{\Gamma }$ point for $k_{0}d=0.5$. Similar distributions up to a phase modulation are observed at the $M$ point. (c) Zoom of the bound states distribution at $\mathrm{\Gamma }$ and M point for $k_{0}d=0.5\pi ,0.7\pi$. (d) Two-excitation spectrum at the $M$ point (upper panel) and average relative distance between two excitations, $⟨r⟩$, (lower panel) as a function of $k_{0}d$. The continuous lines indicate the BSs while the dashed ones in the gray area indicate the scattering resonances branches.

Figure 3

(a) Decay rates ${\gamma }_s^{(2)}$ of the interacting states as a function of $k_{0}d$ for an array of $N=10\times 10$ atoms. (b) Finite size scaling of the decay rates ${\gamma }_s$ with the system size $N$ in the 2D waveguide case (filled symbols) compared to the free space scenario (empty symbols). Here we fixed $k_{0}d=0.52\pi$ for the 2D waveguide while $k_{0}d=1.09\pi$ and $k_{0}d=0.73\pi$ for the $\mathrm{\Gamma }$ and $M$ point in the free space case. In both plots, we considered as repulsive state the type II one.

Figure 4

(a) Two-excitation probability distribution in the relative (left panel) and lab coordinate (right panel) space at times $\gamma t=0$, 20, 300 for $\ell =1$ and $k_{0}d=0.52\pi$. (b) Time evolution of the two-particle correlator, $C_{\mathit{\ell }}(t)$, for different interatomic distances $k_{0}d$ and different initial populations ($\ell =1$, 2). For all figures we fixed $N=10\times 10$.