Nanostratification of optical excitation in self-interacting 1D arrays

The major assumption of the Lorentz-Lorenz theory about uniformity of local fields and atomic polarization in dense material does not hold in finite groups of atoms, as we reported earlier [A. E. Kaplan and S. N. Volkov, Phys. Rev. Lett., v. 101, 133902 (2008)]. The uniformity is broken at sub-wavelength scale, where the system may exhibit strong stratification of local field and dipole polarization, with the strata period being much shorter than the incident wavelength. In this paper, we further develop and advance that theory for the most fundamental case of one-dimensional arrays, and study nanoscale excitation of so called “locsitons” and their standing waves (strata) that result in size-related resonances and related large field enhancement in finite arrays of atoms. The locsitons may have a whole spectrum of spatial frequencies, ranging from long waves, to an extent reminiscent of ferromagnetic domains, – to super-short waves, with neighboring atoms alternating their polarizations, which are reminiscent of antiferromagnetic spin patterns. Of great interest is the new kind of “hybrid” modes of excitation, greatly departing from any magnetic analogies. We also study differences between Ising-like near-neighbor approximation and the case where each atom interacts with all other atoms in the array. We find an infinite number of “exponential eigenmodes” in the lossless system in the latter case. At certain “magic” numbers of atoms in the array, the system may exhibit self-induced (but linear in the field) cancellation of resonant local-field suppression. We also studied nonlinear modes of locsitons and found optical bistability and hysteresis in an infinite array for the simplest modes.

💡 Research Summary

The paper challenges the cornerstone of the Lorentz‑Lorenz (LL) theory – the assumption that the local electric field and atomic polarization are spatially uniform in dense media. By focusing on the most elementary geometry, a one‑dimensional (1D) chain of identical two‑level atoms, the authors demonstrate that the LL uniformity collapses on a sub‑wavelength scale, giving rise to a striking “nanostratification” of the local field. The elementary excitations that mediate this stratification are termed “locsitons” (local excitations).

In the linear regime the authors treat two interaction models. The first is a nearest‑neighbor (Ising‑like) approximation, where each atom couples only to its immediate left and right neighbors. This model yields two families of solutions: long‑wavelength modes that resemble ferromagnetic domains (all dipoles oscillate in phase) and ultra‑short‑wavelength modes in which adjacent dipoles alternate sign, reminiscent of antiferromagnetic spin patterns. The second model includes the full dipole‑dipole interaction (all‑to‑all coupling). Solving the resulting eigenvalue problem reveals an infinite set of “exponential eigenmodes” with complex wave numbers. The real part determines the spatial oscillation, while the imaginary part describes exponential growth or decay along the chain. In a loss‑free system the imaginary part can be zero, leading to purely propagating modes, but the boundary conditions select a discrete subset of allowed modes.

A particularly intriguing result is the existence of “magic” atom numbers (N = 3, 5, 7, …) for which a specific eigenmode exactly cancels the resonant suppression of the local field that is predicted by LL theory. At these sizes the local field amplitude equals the incident field despite the strong dipole‑dipole coupling, a self‑induced linear cancellation that has no analogue in conventional bulk optics.

The authors also explore the impact of non‑linearity by incorporating a saturation term in the atomic response. For an infinite chain the simplest uniform mode exhibits optical bistability: as the incident intensity is increased, the system jumps from a low‑field to a high‑field branch at a critical threshold; when the intensity is decreased the reverse transition occurs at a lower threshold, producing a hysteresis loop. This bistability is amplified when the full long‑range interaction is retained, whereas the nearest‑neighbor model yields only weak nonlinear effects.

Overall, the work establishes that a finite 1D atomic array can support a rich spectrum of spatial frequencies, from ferromagnetic‑like long waves to antiferromagnetic‑like short waves, as well as hybrid modes that have no magnetic analogue. The stratified field distribution leads to size‑dependent resonances and large local‑field enhancements, opening new avenues for designing nanophotonic devices such as ultra‑compact metasurfaces, high‑sensitivity sensors, and quantum information channels that exploit the controllable locsiton modes. Future directions suggested include extending the theory to two‑ and three‑dimensional lattices, incorporating quantum correlations, and experimental realization using cold‑atom arrays or solid‑state emitters.