Band Structure and Dynamics of Single Photons in Atomic Lattices

TL;DR

The paper tackles how dimensionality governs single-photon band structure and cooperative decay in infinite atomic lattices embedded in 3D free space. It develops a real-space, regularization-free framework using a theta-function transform and Ewald summation to compute lattice sums and derive a universal pole equation for the band structure in 1D, 2D, and 3D. Key findings include complex, radiative bands with gaps and oscillatory decay in 1D/2D, contrasted with purely real, non-radiative bands in 3D except at Bragg resonances, and dynamics transitioning from dissipative decay to coherent transport with dimensionality. The results illuminate how lattice geometry and Bragg conditions control light–matter cooperativity, with implications for subradiant states, photonic band gaps, and quantum networking designs.

Abstract

We present a framework to investigate the collective properties of atomic lattices in one, two, and three dimensions. We analyze the single-photon band structure and associated atomic decay rates, revealing a fundamental dependence on dimensionality. One- and two-dimensional arrays are shown to be inherently radiative, exhibiting band gaps and decay rates that oscillate between superradiant and subradiant regimes, as a function of lattice spacing. In contrast, three-dimensional lattices are found to be fundamentally non-radiative due to the inhibition of spontaneous emission, with decay only at discrete Bragg resonances. Furthermore, we demonstrate that this structural difference dictates the system dynamics, which crosses over from dissipative decay in lower dimensions to coherent transport in three dimensions. Our results provide insight into cooperative effects in atomic arrays at the single-photon level.

Figures (9)

• Figure 1: Band structure of a 1D atomic lattice. Solid lines show the numerical solution for the lower (subradiant) and upper (superradiant)branches of the band structure, while dashed lines show the result from the pole approximation (Eq. \ref{['eq:pole approx']}). (a) Real part of the complex energy $\alpha(\beta)$, corresponding to the collective frequency shift. (b) Imaginary part of the energy, representing the collective decay rate $\Gamma(\beta)$. The horizontal red dashed line indicates the single-atom spontaneous emission rate $\Gamma_0$ for comparison. We set $\alpha_{0} = 0.30$ and $\kappa = 5 \times 10^{-3}$ here and throughout.
• Figure 2: Decay rate versus lattice spacing in a 1D lattice. (a) The collective decay rate $\Gamma(\beta)$ (normalized by the single-atom rate $\Gamma_0$) calculated via the pole approximation as a function of the dimensionless lattice spacing $\alpha_0$ for various dimensionless momenta $\beta$. The curves display strong oscillations alternating between subradiant ($\Gamma/\Gamma_0<1$) and superradiant ($\Gamma/\Gamma_0>1$) behavior. (b) Schematic of the first Brillouin zone illustrating the light-cone boundary $|\beta|=\alpha_0$: modes are non-radiative for $|\beta|>\alpha_0$ and become radiative when $\alpha_0>| \beta|$. The vertical dashed lines in (a) indicate the critical spacings $\alpha_0=|\beta|$ where each mode crosses the light-cone and radiative decay is activated.
• Figure 3: Illustrating the dynamics in a 1D lattice. (a) Temporal evolution of the atomic population $|\psi_n(t)|^2$ for the initially excited atom ($n=0$) and its nearest neighbors ($n=1, 2$), showing excitation transfer and decay with dimensionless time $\tau = c t / a$. (b) The spatial profile of the excitation probability $|\psi_n(t)|^{2}$ at different time snapshots, illustrating the formation of a localized, non-propagating wavepacket.
• Figure 4: Band structure of a 2D square atomic lattice. (a) Real part of the complex energy, showing the polaritonic dispersion. The shaded region highlights the band gap. (b) Imaginary part of the energy, representing the collective decay rate. Note the significant superradiant enhancement at the high-symmetry points.
• Figure 5: Decay rate versus lattice spacing in a 2D square lattice. (a) Collective decay rate $\Gamma(\bm{\beta})$ as a function of the dimensionless lattice spacing $\alpha_0=a/\lambda_0$ for the high-symmetry points $\Gamma$, X, and M, where $\lambda_0=$. The vertical dashed lines indicate the critical thresholds $\alpha_0 = 1/2$ and $\alpha_0 = \sqrt{2}/2$, below which the X and M points, respectively, become non-radiative dark states. (b) Illustration of the light cone in the first Brillouin zone. Modes located inside the circular region, defined by the boundary $\alpha(\bm{\beta}) = \sqrt{\beta_{1}^{2}+\beta_{2}^{2}}$, are radiative, while those outside are non-radiative.