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

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 $ξ\sim Δ^{-1/d_w}$, with the detuning $Δ$ 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 Sierpiński gaskets, pyramids, Vicsek graphs, and Sierpiński 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 Sierpiński 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.

• The paper establishes that the localization length scales as ξ ∼ Δ^(-1/d_w), deviating from the traditional Δ^(-1/2) law observed in periodic systems.
• It employs a heat kernel representation and exact diagonalization to connect the anomalous diffusion exponent d_w with the fractal geometry of the photonic bath.
• The findings highlight that engineering fractal photonic baths can provide a novel regime for controlling quantum light-matter interactions.

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

Introduction

The study "Atom-Photon Bound States in Fractal Photonic Lattices: Localization Length and Anomalous Diffusion" (2605.23625) provides a comprehensive analysis of single-emitter bound states in self-similar, fractal photonic lattices. In contrast to conventional periodic systems, fractal photonic lattices lack translational invariance, have non-integer Hausdorff dimension, and support anomalous diffusion. The paper establishes a connection between the localization properties of atom-photon bound states and the fundamental transport exponents of the underlying fractal geometry, principally the walk dimension $d_w$. The study employs analytic arguments based on the heat kernel representation of the bath Green's function and validates key predictions through exact diagonalization on canonical fractal graphs.

Figure 1: Illustration of atom-photon bound states for emitters in regular and fractal lattices; in regular arrays, $\xi\sim\Delta^{-1/2}$ due to quadratic band edge dispersion, while in fractals, the localization exponent is controlled by the walk dimension as $\xi\sim \Delta^{-1/d_w}$ ($d_w \approx 2.32$ for Sierpiński gaskets).

From Periodic to Fractal Photonic Baths

Atom-photon bound states in periodic photonic baths are understood within band theory; localization length $\xi$ diverges near the photonic band edge with $\xi\sim\Delta^{-1/2}$, where $\Delta$ describes the detuning from the lower edge and the scaling exponent is set by the quadratic dispersion ($d_w=2$) near the band edge. This effective mass picture is invalidated for fractal lattices, where momentum space is ill-defined due to the absence of translation symmetry, and exciton- or photon transport exhibits subdiffusive behavior.

Key distinctions arise in fractals, such as Sierpiński gaskets and carpets:

• Finitely ramified (nested) fractals: Removable by excising a finite set of sites at any scale.
• Infinitely ramified fractals: Remain globally connected regardless of finite removals, e.g., Sierpiński carpets.

Self-similarity and varying local connectivity introduce new transport exponents beyond the fractal (Hausdorff) dimension. Notably, the walk dimension $d_w$ (characterizing anomalous diffusion: $\langle r^2(t)\rangle \sim t^{2/d_w}$) and the spectral dimension $\xi\sim\Delta^{-1/2}$0 (controlling the density of states and return probability scaling).

Figure 2: Generational construction of nested finitely ramified (a) and infinitely ramified (b) self-similar photonic graphs, such as Sierpiński gaskets and carpets.

Analytical Framework: Green's Function via Heat Kernel

The crucial analytic step is expressing the photonic Green's function at the bound state energy as a Laplace transform of the heat kernel, $\xi\sim\Delta^{-1/2}$1, for the underlying graph. This enables leveraging known rigorous bounds for diffusion on fractals. The far-field asymptotic of the bound state is then controlled by the heat-kernel's long-time and large-distance decay:

$\xi\sim\Delta^{-1/2}$2

The principal result is that the localization length diverges as

$\xi\sim\Delta^{-1/2}$3

in the limit of vanishing detuning, where $\xi\sim\Delta^{-1/2}$4 is the random-walk dimension specific to the fractal. This scaling law persists universally for all Laplacian-like Hamiltonians on self-similar graphs, in contrast to the effective mass law in periodic systems.

For the near-field (short-distance) regime, the bound state displays additional algebraic corrections controlled by $\xi\sim\Delta^{-1/2}$5 ($\xi\sim\Delta^{-1/2}$6 being the fractal dimension), leading to

$\xi\sim\Delta^{-1/2}$7

This scaling aligns with first-passage and resistance scaling in classical transport for finitely ramified fractals, while deviations occur for infinitely ramified structures.

Numerical Results: Validation and Characterization

The analytic predictions are systematically validated via exact diagonalization for a suite of fractal and reference lattices, employing boundary-coupled emitters to resolve localization directly from the asymptotic exponential decay.

• The localization length for all analyzed fractals accurately follows $\xi\sim\Delta^{-1/2}$8, in agreement with theoretical $\xi\sim\Delta^{-1/2}$9 values (e.g., $\xi\sim \Delta^{-1/d_w}$0 for Sierpiński gasket $\xi\sim \Delta^{-1/d_w}$1), demonstrating that the transport exponent governs the tail behavior of the bound state.

  Figure 3: The numerical localization length, rescaled as $\xi\sim \Delta^{-1/d_w}$2, reveals systematic deviation from flatness (periodic-lattice benchmark), confirming the fractal power law $\xi\sim \Delta^{-1/d_w}$3.

• Near-field analysis exposes that for nested, finitely ramified fractals, the amplitude difference displays power-law scaling as predicted by resistance exponent $\xi\sim \Delta^{-1/d_w}$4, whereas infinitely ramified structures, e.g., Sierpiński carpets, deviate and align more closely with the marginal two-dimensional logarithmic case.

  Figure 4: Near-field scaling of amplitude differences, showing algebraic scaling for nested fractals in quantitative agreement with theoretical $\xi\sim \Delta^{-1/d_w}$5, and systematic deviations for Sierpiński carpets.

Implications and Future Directions

This study articulates that, in structured baths with fractal geometry, the relevant control parameters for atom-photon bound state localization are no longer band curvature or effective mass but the anomalous transport exponents $\xi\sim \Delta^{-1/d_w}$6 (and to a lesser extent $\xi\sim \Delta^{-1/d_w}$7, $\xi\sim \Delta^{-1/d_w}$8). This insight substantially broadens the design space for quantum photonic environments: By engineering a fractal bath, one can control the spatial extent and decay profiles of light-matter bound states in a manner inaccessible to periodic band-structure engineering.

Potential future research directions include:

• Analysis of many-emitter interactions, effective spin models, and non-Gaussian photonic baths in fractal geometries.
• Exploring disorder and decoherence effects atop self-similar backgrounds.
• Experimental realization with femtosecond-laser-written waveguide arrays or superconducting circuits engineered to support fractal connectivity patterns.
• Generalization to other non-periodic or hyperbolic bath structures, and exploration of topologically protected states in these settings.

The methodology using real-space heat kernel analysis circumvents limitations posed by the lack of translational invariance and should be extensible to other complex graph-based quantum systems.

Conclusion

The findings of (2605.23625) demonstrate that the localization properties of single-excitation atom-photon bound states in fractal photonic lattices are tightly governed by the anomalous diffusion exponent $\xi\sim \Delta^{-1/d_w}$9 intrinsic to the self-similar geometry. The localization length scales as $d_w \approx 2.32$0, in contrast to the standard $d_w \approx 2.32$1 law in periodic systems, and the near-field profile encodes further signatures of the geometry, substantiated by both analytical and numerical evidence. This work positions fractal photonic environments as a distinct and tunable regime for quantum light-matter interaction, with implications for the design of long-range, coherent quantum emitters, and the programmable shaping of electromagnetic environment at the quantum level.