Subradiant Decay in 2D and 3D Atomic Arrays

TL;DR

The paper tackles subradiant decay in 2D and 3D regular atomic arrays, addressing how finite size modifies lifetimes and mode structure. It develops an angular-integral representation of the coupled-dipole problem and uses generalized Bloch (Dicke) states to derive explicit expressions for the cooperative decay rate $\Gamma(\mathbf{k})$ in both infinite and large finite lattices, via the structure factor $F(\mathbf{k})$. Key results include a 2D subradiant landscape with dark states inside a Brillouin-circle and a 3D shell-like pattern of subradiant modes, along with quantitative finite-size scalings such as $\Gamma \sim 1/N_x$ or $\sim 1/N^2$ in different regimes. These findings advance predictive control of light-matter cooperativity, with potential applications to quantum memories, photon storage, and topological photonics, and suggest experimental routes to selectively populate subradiant modes and explore multi-excitation extensions.

Abstract

Subradiance is a phenomenon where coupled emitters radiate light at a slower rate than independent ones. While its observation was first reported in disordered cold atom clouds, ordered subwavelength arrays of emitters have emerged as promising platforms to design highly cooperative optical properties based on dipolar interactions. In this work we characterize the eigenmodes of 2D and 3D regular arrays, using a method which can be used for both infinite and very large systems. In particular, we show how finite-size effects impact the lifetimes of these large arrays. Our results may have interesting applications for quantum memories and topological effects in ordered atomic arrays.

Figures (5)

• Figure 1: Collective decay rate $\Gamma/\Gamma_0$ in the $\mathbf{k}=(k_xd, k_yd)$ plane for the infinite array with (a) $d=\lambda_0/5$ and (b) $d=\lambda_0$, and dipoles oriented along the $x$-axis.
• Figure 2: $\Gamma(0,0)/\Gamma_0$ vs $k_0d$ for a square $N\times N$ lattice with $N=10$ and polarization (a) perpendicular to the plane, $\mathbf{\hat{d}}=(0,0,1)$, and (b) within the atomic plane, $\mathbf{\hat{d}}=(1,0,0)$. The dashed red lines are the solution for an infinite lattice, provided by Eq. \ref{['Gamma:2D:infinite']}). In the case (a), $\Gamma(0,0)=0$ when $0<k_0d<\pi$ for an infinite lattice.
• Figure 3: Collective decay rate $\Gamma(k_x,0)/\Gamma_0$ as a function of $k_xd$, for a square array with $d=0.8\lambda_0$ and $N_x=10$, with a polarization orthogonal to the array ($\mathbf{\hat{d}}=(0,0,1)$). Full blue line: exact solution (\ref{['Gamma:2D:1']}); Red dashed line: approximate solution (\ref{['Gamma:2D:approx']}); Dash-dotted line: solution for the infinite chain (\ref{['Gamma:2D:infinite']}). The vertical dotted line stands for the value $k_x=k_0$.
• Figure 4: Collective decay rate $\Gamma(k_\perp)/\Gamma_0$ as a function (a) of $k_\perp/k_0$ for a square array and $N=10$ (blue line), $N=50$ (red line) and $N=100$ (black line), and (b) of $N$ for $k_\perp/k_0=1.2$ (pink boxes), $1.5$ (red triangles) and $2$ (black circles); the dashed line in (b) shows the $1/N^{2}$ dependence. All simulations realized for $d=\lambda_0/4$.
• Figure 5: Collective decay rate $\Gamma(k_x,0,0)/\Gamma_0$ as a function of $k_xd$ for a cubic array with step $d=\lambda_0/4$ and size $N_x=N_y=N_z=20$, as calculated from the exact expression (\ref{['Gamma:3D']}) (full black line) and from the approximated expression (\ref{['Gamma:3D:approx']}) (red dashed line).