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arxiv: 2507.06308 · v4 · submitted 2025-07-08 · 🪐 quant-ph · cond-mat.mes-hall· physics.optics

Fibonacci Waveguide Quantum Electrodynamics

Florian B\"onsel , Flore K. Kunst , Federico Roccati This is my paper

Pith reviewed 2026-05-19 05:47 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hallphysics.optics
keywords waveguide QEDFibonacci sequenceaperiodic structuresquantum emittersbound stateseffective Hamiltonianmultifractal spectrumSu-Schrieffer-Heeger model
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The pith

Fibonacci-structured photonic waveguides enable decoherence-free interactions between quantum emitters by passing the aperiodic sequence into effective atomic Hamiltonians.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates waveguide quantum electrodynamics in one-dimensional photonic arrays where hopping strengths follow the deterministic Fibonacci-Lucas substitution rule instead of periodic arrangements. This creates structures with a singular continuous energy spectrum and critical eigenstates that sit between ordered and disordered systems. It shows that decoherence-free coherent interactions can be engineered in this setting. For giant emitters resonantly coupled to the basic aperiodic waveguide, atom-photon bound states appear only at specific coupling strengths set by the substitution rule, so the resulting effective atomic Hamiltonian directly inherits the Fibonacci structure. For emitters coupled off-resonantly to an aperiodic Su-Schrieffer-Heeger version, the bound-state profiles become aperiodically modulated and the effective Hamiltonian acquires multifractal properties.

Core claim

In Fibonacci waveguides whose hopping amplitudes are generated by the deterministic Fibonacci-Lucas substitution rule, decoherence-free coherent interactions between quantum emitters are realized. Giant emitters resonantly coupled to the simplest aperiodic waveguide form atom-photon bound states only for coupling configurations dictated by the aperiodic sequence, yielding an effective atomic Hamiltonian that itself follows the Fibonacci structure. Emitters locally and off-resonantly coupled to the aperiodic Su-Schrieffer-Heeger waveguide produce mediating bound states with aperiodically modulated profiles, so the effective Hamiltonian exhibits multifractal properties.

What carries the argument

The Fibonacci-Lucas substitution rule applied to hopping amplitudes, which generates a singular continuous spectrum and critical eigenstates that dictate the allowed coupling configurations for bound-state formation and thereby imprint the aperiodic structure onto the effective emitter Hamiltonian.

If this is right

  • Effective Hamiltonians for quantum emitters can directly inherit deterministic aperiodic sequences for engineered interactions.
  • Off-resonant coupling to aperiodic Su-Schrieffer-Heeger waveguides produces multifractal effective Hamiltonians.
  • The platform remains experimentally feasible in photonic devices while preserving coherence.
  • Deterministic complexity of aperiodic structures can be transferred into quantum-emitter interactions without disorder.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same substitution-rule approach could be applied to other deterministic aperiodic sequences to obtain different spectral and interaction properties.
  • Multifractal effective Hamiltonians might support novel many-body phases when multiple emitters are coupled together.
  • Fabrication tolerances in real devices could be tested by checking whether small deviations from the ideal substitution rule still preserve the predicted bound-state selectivity.

Load-bearing premise

The deterministic Fibonacci-Lucas substitution rule applied to hopping amplitudes produces bound states only for specific coupling configurations that can be realized in a real photonic device without adding extra decoherence channels.

What would settle it

An experimental measurement in a fabricated one-dimensional photonic array with Fibonacci-Lucas hopping amplitudes that checks whether atom-photon bound states appear exclusively at the coupling strengths predicted by the substitution rule or appear for arbitrary couplings as well.

Figures

Figures reproduced from arXiv: 2507.06308 by Federico Roccati, Flore K. Kunst, Florian B\"onsel.

Figure 1
Figure 1. Figure 1: Standard vs Fibonacci waveguide QED. Quantum emitters [blue] coupled to one-dimensional arrays of resonators (waveguides) [red]. The photon-mediated decoherence-free interactions between emitters arise from virtual photons that are strictly localized between the coupling points of the giant (multilocal) emitters in the resonant case [left], or from localized virtual photonic clouds in the vicinity of off-r… view at source ↗
Figure 2
Figure 2. Figure 2: Standard vs Fibonacci waveguides. In both panels (a) and (b) we show: the density of states, DOS, (top row), and integrated DOS, N (E). Panel (a): uniform waveguide (left) and its aperiodic version, the (1, 1)-Fibonacci waveg￾uide (right). Panel (b): SSH waveguide (left) and its aperiodic version, the (1, 2)-Fibonacci waveguide (right). Notice that a central gap persist (shaded blue) in addition to all the… view at source ↗
Figure 3
Figure 3. Figure 3: Singularity spectrum. For delocalized (localized) models f(α) shrinks to a single point f(1) = 1 shown in red (f(0) = 0, yellow triangle). Multifractal models have a con￾tinuum of scaling exponents. Parameter values: AAH model, V = 1.5 t, t, 0.5 t for the localized, critical, and localized phase, respectively; uniform waveguide, tB = tA; (1, 1)-Fibonacci, (1, 2)-Fibonacci, and SSH waveguides, tB = 0.2tA. A… view at source ↗
Figure 4
Figure 4. Figure 4: Inverse Participation Ratio. Inverse participation ratio averaged over all eigenstates versus system size. For delocalized models (as the uniform and SSH waveguides un￾der periodic boundary conditions, and the AAH model with V < t) the IPR scales with system size ∝ N −1 . For localized models (as the AAH model for V > t) the IPR is constant. Critical models (as the (1, 1)- and (1, 2)-Fibonacci waveguides, … view at source ↗
Figure 5
Figure 5. Figure 5: Giant emitters in a (1, 1)-Fibonacci waveguide. (a) Giant emitters arranged in the first nontrivial configuration yielding decoherence-free interactions (coupling points at distance d = 6). In this configuration, giant atoms can induce three kinds of atom-photon bound states |Ψa⟩, |Ψb⟩ and its mirror symmetric one, whose mutual overlap gives the coherent emitter interactions. On the right we show the block… view at source ↗
Figure 6
Figure 6. Figure 6: Emitters in a (1, 2)-Fibonacci waveguide. (a) Local off-resonant emitters yield chiral atom-photon bound states whose photonic profile decays exponentially with aperiodic modulation. (b) DOS and integrated DOS of the effective emitter Hamiltonian showing its critical behavior. Parameter values: tB = 0.2 tA, g = 0.05 tA. even sites and to the left of the atom. If instead nj is even we get [PITH_FULL_IMAGE:… view at source ↗
Figure 7
Figure 7. Figure 7: Gap size. Size of the central bandgap of (p, q)- Fibonacci waveguides as a function of p and q (the values of p and q not displayed correspond to gapless waveguides). We used tB = 0.2 tA. (a) (b) (g2 /tA)t emitter excitation 0.0 2.5 5.0 (g2 /tA)t 0.0 2.5 5.0 0.0 0.2 0.4 0.6 0.8 1.0 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Exact vs effective dynamics. Dynamics of the emit￾ters’ populations for three emitters starting from the state |e⟩1 (only the first emitter is excited). Dashed lines correspond to exact dynamics, |j ⟨e| e −iHt ˆ |e⟩ j | 2 , while solid lines to effective dynamics, |j ⟨e| e −iHˆ efft |e⟩ j | 2 . First, second, and third emit￾ters correspond to red, turquoise and blue. (a): Three local atoms coupled at n0 = … view at source ↗
read the original abstract

Waveguide quantum electrodynamics (QED) provides a powerful framework for engineering quantum interactions, traditionally relying on periodic photonic arrays with continuous energy bands. Here, we investigate waveguide QED in a fundamentally different environment: A one-dimensional photonic array whose hopping strengths are structured aperiodically according to the deterministic Fibonacci-Lucas substitution rule. These "Fibonacci waveguides" lack translational invariance and are characterized by a singular continuous energy spectrum and critical eigenstates, representing a deterministic intermediate between ordered and disordered systems. We demonstrate how to achieve decoherence-free, coherent interactions in this unique setting. We analyze two paradigmatic cases: (i) Giant emitters resonantly coupled to the simplest aperiodic version of a standard waveguide. For these, we show that atom photon bound states form only for specific coupling configurations dictated by the aperiodic sequence, leading to an effective atomic Hamiltonian, which itself inherits the Fibonacci structure; and (ii) emitters locally and off-resonantly coupled to the aperiodic version of the Su-Schrieffer-Heeger waveguide. In this case the mediating bound states feature aperiodically modulated profiles, resulting in an effective Hamiltonian with multifractal properties. Our work establishes Fibonacci waveguides as a versatile platform, which is experimentally feasible, demonstrating that the deterministic complexity of aperiodic structures can be directly engineered into the interactions between quantum emitters.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript explores waveguide quantum electrodynamics in aperiodic Fibonacci waveguides defined by the Fibonacci-Lucas substitution rule for hopping amplitudes. It analyzes two cases: giant emitters coupled to the basic aperiodic waveguide, where bound states form only for sequence-specific couplings leading to Fibonacci-structured effective atomic Hamiltonians; and local off-resonant coupling to an aperiodic Su-Schrieffer-Heeger waveguide, yielding multifractal effective Hamiltonians. The central thesis is that such structures enable decoherence-free coherent interactions by inheriting the aperiodic properties.

Significance. This research holds potential significance in quantum optics and condensed matter physics by demonstrating how deterministic aperiodic order can be transferred to quantum emitter interactions. If the derivations are rigorous, it opens avenues for designing complex quantum gates or simulators using photonic platforms with critical eigenstates. The emphasis on experimental feasibility adds practical value, distinguishing it from purely theoretical aperiodic models.

major comments (2)
  1. [Abstract and giant emitters case] Abstract and analysis of giant emitters: The claim that atom-photon bound states form only for specific coupling configurations dictated by the aperiodic sequence is central but lacks an explicit derivation showing how the standard resolvent or pole-search method is adapted to the singular continuous Cantor spectrum of the Fibonacci waveguide. A concrete recursion relation or transfer-matrix condition would be required to establish this specificity.
  2. [SSH waveguide case] Analysis of off-resonant SSH case: The resulting effective Hamiltonian is asserted to have multifractal properties due to aperiodically modulated bound-state profiles, but this requires quantitative support such as a computed fractal dimension or scaling exponent rather than qualitative description to substantiate the inheritance of structure.
minor comments (2)
  1. [Model definition] The exact form of the Fibonacci-Lucas substitution rule for the hopping amplitudes should be stated explicitly in the model definition to allow reproducibility.
  2. [Figures] Any figures showing bound-state wavefunctions or effective couplings should include direct comparison to the periodic limit to highlight the aperiodic effects.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and have revised the manuscript to provide the requested clarifications and quantitative support.

read point-by-point responses

  1. Referee: [Abstract and giant emitters case] Abstract and analysis of giant emitters: The claim that atom-photon bound states form only for specific coupling configurations dictated by the aperiodic sequence is central but lacks an explicit derivation showing how the standard resolvent or pole-search method is adapted to the singular continuous Cantor spectrum of the Fibonacci waveguide. A concrete recursion relation or transfer-matrix condition would be required to establish this specificity.

    Authors: We appreciate the referee's point on the need for explicit detail. Our analysis adapts the resolvent formalism to the Fibonacci waveguide's spectrum, but we agree an expanded derivation would improve clarity. In the revised manuscript we have added a dedicated subsection deriving the recursion relation from the transfer-matrix method for the singular continuous spectrum. This explicitly shows that bound-state poles exist only for coupling configurations matching the Fibonacci sequence, establishing the required specificity without altering the original conclusions. revision: yes

  2. Referee: [SSH waveguide case] Analysis of off-resonant SSH case: The resulting effective Hamiltonian is asserted to have multifractal properties due to aperiodically modulated bound-state profiles, but this requires quantitative support such as a computed fractal dimension or scaling exponent rather than qualitative description to substantiate the inheritance of structure.

    Authors: We agree that quantitative measures provide stronger substantiation. In the revised manuscript we have included computations of the fractal dimension (via box-counting) of the effective Hamiltonian spectrum and the scaling exponents of the bound-state profiles. These results, presented in a new figure with accompanying analysis, confirm the multifractal character and its direct inheritance from the aperiodic waveguide modulation. revision: yes

Circularity Check

0 steps flagged

Derivation of effective Hamiltonians remains self-contained from aperiodic substitution rule

full rationale

The paper starts from the waveguide Hamiltonian with hopping amplitudes defined by the deterministic Fibonacci-Lucas substitution rule, then analyzes bound-state formation for giant emitters and off-resonant SSH cases using adapted Green's function or transfer-matrix methods suited to the singular-continuous spectrum. The resulting effective atomic Hamiltonians are obtained by explicit computation of mediated interactions that inherit the aperiodic modulation; no parameters are fitted to match target observables, no load-bearing uniqueness theorem is imported via self-citation, and the inheritance of Fibonacci structure follows directly from the input lattice definition rather than being presupposed. The derivation supplies independent content by showing which specific coupling configurations permit bound states without additional decoherence, making the central claims falsifiable against the underlying photonic model.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard tight-binding description of photonic waveguides, the definition of the Fibonacci-Lucas substitution rule, and the existence of atom-photon bound states in the resulting non-periodic lattice; no new free parameters or invented particles are introduced in the abstract.

axioms (2)
  • domain assumption Photonic array described by nearest-neighbor tight-binding Hamiltonian with position-dependent hoppings
    Standard modeling choice in waveguide QED invoked to define the Fibonacci structure.
  • domain assumption Emitters couple locally to the photonic modes and form bound states when detuned or resonant
    Core assumption enabling the effective atomic Hamiltonian construction.

pith-pipeline@v0.9.0 · 5777 in / 1356 out tokens · 50220 ms · 2026-05-19T05:47:49.706566+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith.Constants phi_golden_ratio echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    hopping strengths are structured aperiodically according to the deterministic Fibonacci-Lucas substitution rule... atom photon bound states form only for specific coupling configurations dictated by the aperiodic sequence, leading to an effective atomic Hamiltonian, which itself inherits the Fibonacci structure

  • IndisputableMonolith.Foundation.AlexanderDuality alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    singular continuous energy spectrum and critical eigenstates... multifractal properties

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Forward citations

Cited by 1 Pith paper

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Reference graph

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