Unsplit Spreading: An Overlooked Signature of Long-Range Interaction

TL;DR

This work identifies unsplit spreading as a smoking-gun signature of singular band structure caused by long-range interactions. It proves a no-go theorem: for a lattice with a smooth dispersion $\omega(k)$, an initially localized excitation must split into two counter-propagating wave packets, with the split arising from zeros of $\partial_k^2\omega(k)$ and, in 2D, from a Gauss–Bonnet topological constraint. Long-range couplings induce singular features in $\omega(k)$ that circumvent this constraint, enabling unsplit spreading in 1D and 2D systems, including subwavelength atom arrays in waveguide QED and free space, often within subradiant sectors of open systems. The results provide a measurable diagnostic for long-range physics and motivate further many-body studies of dynamics under singular dispersions.

Abstract

In conventional lattice models, the dispersion relation $ω(k)$ is assumed to be a smooth function. We prove that this smoothness implies the splitting of an initially localized excitation into counter-propagating wave packets. Consequently, unsplit spreading can occur only when $ω(k)$ develops singular features, precisely what long-range interactions enable. Remarkably, this phenomenon was clearly visible in published quantum simulation experiments as early as 2014, yet it has remained unrecognized or discussed as a distinct physical effect. We show that unsplit spreading emerges in realistic open quantum systems, such as 1D and 2D subwavelength atomic arrays, where the long-lived subradiant states host effective dispersion with the required singularities. Our work establishes unsplit spreading as an experimentally accessible, smoking-gun signature of singular band structure induced by long-range physics.

Figures (4)

• Figure 1: An initially localized excitation on a lattice exhibits two distinct diffusion behaviors: (a) it splits into two broadening wave packets that propagate in opposite directions; (b) it spreads without splitting. The solid curves represent the envelopes of the waveforms. As we will demonstrate, behavior (b) is a signature of singularities in the system’s dispersion relation arising from long-range interactions.
• Figure 2: Spreading of waveforms under long-range interactions decaying as $1/r^\alpha$, with (a)–(c) corresponding to $\alpha=1, 2$ and $3$, respectively. For each case, the waveforms at three different times are shown. Time is measured in units of $t_0=a^{\alpha}$ where $a$ is the lattice constant.
• Figure 3: Left panels: $\partial_k^2\Re\omega(k)$, the second derivative of the real part of the dispersion relation. Right panels: snapshots of the waveform of an atomic excitation initialized at the center of a chain of 751 atoms. Panels (a)–(c) correspond to (a) waveguide QED and (b, c) free space, with atomic dipole $\bm{d}$ polarized parallel and perpendicular to the chain, respectively. The first Brillouin Zone is shifted to $[-k_A, 2\pi-k_A]$ for visual convenience. The probability that the excitation survives decaying is indicated near each snapshot.
• Figure 4: Waveforms of an excitation initialized in the center of a 2D square lattice ($93\times93$) with (a) $k_A =0.3\pi$; (b) $k_A =1.2\pi$. The atomic dipoles are polarized along a fixed direction $\sin(\pi/12)/\sqrt{2}(\hat{\bm{x}}+\hat{\bm{y}})+ \cos(\pi/12)\hat{\bm{z}}$, where $\hat{\bm{x}}$ and $\hat{\bm{y}}$ are unit vectors along the lattice primitive directions and $\hat{\bm{z}}$ is normal to the lattice plane. "Population" refers to the survival probability of the excited state against spontaneous emission.