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arXiv · 2605.23625

Atom-Photon Bound States in Fractal Photonic Lattices: Localization Length and Anomalous Diffusion

Florian Bönsel·Flore K. Kunst·Federico Roccati
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Abstract

We study atom-photon bound states seeded by two-level emitters coupled to self-similar photonic lattices. By expressing the photonic Green's function through the heat kernel, we show that the far-field localization length obeys $\xi \sim \Delta^{-1/d_w}$, with the detuning $\Delta$ from the lower spectral edge and the walk dimension $d_w$ of the underlying fractal. This scaling is controlled by anomalous diffusion and does not rely on translational invariance or a band-edge effective-mass approximation. Exact diagonalization on Sierpi\'nski gaskets, pyramids, Vicsek graphs, and Sierpi\'nski carpets confirms the far-field prediction once the bath Hamiltonian is rendered Laplacian-like by compensating the local inhomogeneity in the connectivities with on-site potentials. In the near field, the bound-state amplitude exhibits an additional algebraic variation. For nested finitely ramified fractals, the corresponding exponent agrees with the classical resistance/ first-passage scaling, whereas Sierpi\'nski carpets display clear deviations from this simple law. Our results extend structured-bath waveguide QED to self-similar non-periodic geometries and connect bound-state profiles to transport exponents of the underlying fractal lattice.

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Florian Bönsel, Flore K. Kunst, Federico Roccati. 2026-05-22. Atom-Photon Bound States in Fractal Photonic Lattices: Localization Length and Anomalous Diffusion. https://arxiv.org/abs/2605.23625

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