Band Structure and Dynamics of Single Photons in Atomic Lattices

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.

💡 Research Summary

This paper develops a unified, regularization‑free theoretical framework for calculating the single‑photon band structure and collective decay rates of infinite atomic lattices in one, two, and three dimensions. The authors start from a scalar quantum‑optics Hamiltonian describing two‑level atoms coupled to the three‑dimensional free‑space electromagnetic field, restrict the dynamics to the single‑excitation subspace, and eliminate the photonic degrees of freedom via a Laplace transform. The resulting effective atomic equation contains a lattice sum S^{(d)}(α,β) that encodes the long‑range, photon‑mediated dipole‑dipole interaction.

Instead of the conventional momentum‑space Poisson summation, which requires an artificial cutoff to tame the divergent self‑energy (Lamb shift), the authors evaluate the lattice sum directly in real space using a combination of the theta‑function transform and Ewald summation. This approach removes the on‑site self‑energy term entirely and yields rapidly convergent expressions for S^{(d)} in any dimension. The complex band energies α(β) are defined as the roots of a transcendental equation involving S^{(d)}; the real part gives the collective frequency shift, while the imaginary part gives the collective decay rate.

In one dimension, the lattice sum can be expressed analytically in terms of Bernoulli polynomials, Gamma functions, and logarithms. Numerical solutions reveal two distinct branches: an upper super‑radiant branch with large decay rates and a lower sub‑radiant branch that can become completely dark when the Bloch wavevector lies outside the light cone. A first‑order pole approximation α≈α₀−2πiκα₀²−κS^{(1)}(α₀,β) reproduces the full solution near resonance. The decay rate Γ(β) oscillates with lattice spacing a (through the dimensionless parameter α₀=Ωa/2πc), alternating between super‑radiant and sub‑radiant regimes. In the limit a→0 the zero‑momentum mode becomes infinitely super‑radiant, while all other modes become sub‑radiant; for a→∞ the decay approaches the single‑atom spontaneous emission rate. Dynamical simulations, starting from an excitation localized on a single atom, show rapid exponential decay and short‑range energy transfer to neighboring sites, confirming the dissipative nature of 1D lattices.

Two‑dimensional square lattices exhibit qualitatively similar behavior but with richer interference patterns due to the extra degree of freedom. The band structure remains complex, featuring a sizable band gap and direction‑dependent super‑radiant peaks. The collective decay rates again show non‑monotonic dependence on spacing, with broader sub‑radiant regions because many Bloch vectors lie outside the light cone.

In stark contrast, the three‑dimensional simple‑cubic lattice is fundamentally non‑radiative. The real‑valued band structure indicates that, except at discrete Bragg resonances where the Bloch vector satisfies a reciprocal‑lattice condition, the imaginary part of α vanishes. Consequently, an initially localized excitation persists indefinitely, demonstrating coherent transport without dissipative loss. This result confirms earlier predictions of spontaneous‑emission inhibition in fully three‑dimensional photonic band‑gap structures, now derived from an atom‑level microscopic model without any cutoff.

The paper validates its analytical formulas against existing numerical studies, reproducing previously reported oscillations of decay rates and confirming the accuracy of the pole approximation. By eliminating the need for artificial regularization, the authors provide a versatile tool that can be extended to vector electromagnetic fields, other lattice geometries, and multi‑photon sectors. The dimensional dependence uncovered here—radiative decay in 1D/2D versus lossless propagation in 3D—has direct implications for designing quantum‑optical devices such as quantum memories, sub‑radiant metasurfaces, and directional photon routers that exploit cooperative effects at the single‑photon level.