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arxiv: 2510.22468 · v1 · submitted 2025-10-26 · 🪐 quant-ph

Single-photon superradiance and subradiance in helical collectives of quantum emitters

Hamza Patwa , Philip Kurian This is my paper

Pith reviewed 2026-05-18 04:56 UTC · model grok-4.3

classification 🪐 quant-ph
keywords superradiancesubradiancecollective decayLamb shifthelical emitterstwo-level systemsprotein fiberscontinuous distribution
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The pith

Continuous distributions of two-level systems on an infinite helix yield closed-form expressions for collective decay rates and Lamb shifts under single-photon excitation.

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

The paper derives exact analytical expressions for the collective decay rates and Lamb shifts that arise when a single photon interacts with two-level emitters distributed continuously along an infinite straight line or an infinite helix. These formulas are obtained by solving the eigenvalue problem for the interaction in the continuous limit and are then compared against cylindrical geometries, discrete emitter lattices, and interactions truncated at different ranges. A reader would care because the closed forms make it possible to calculate the fraction of superradiant, subradiant, and trapped states without diagonalizing large matrices, and the helix solution is applied directly to estimate behavior in protein-fiber architectures where helical geometry is native. The work also identifies the conditions under which the continuous scalar model agrees with or departs from discrete vector treatments.

Core claim

Novel analytical expressions are obtained for the collective decay rates and Lamb shifts of a single photon coupled to a continuous distribution of two-level systems placed along an infinite line and along an infinite helix. These expressions are compared with the eigenvalues for cylindrical arrangements, revealing dimensional limits in which the spectra coincide. Direct comparison with discrete lattices shows that the continuous scalar model does not recover the discrete vector results even in the zero-spacing limit, and that short-, intermediate-, and long-range interaction terms produce quantitatively different spectra. The helix formulas are then used to estimate the maximally superradiy

What carries the argument

The closed-form eigenvalue spectrum of the continuous interaction kernel for an infinite helical chain of two-level systems.

If this is right

  • The fraction of trapped states and the thermally averaged decay rate for helical protein-fiber architectures can be read off directly from the analytical spectrum without large-scale numerics.
  • Limits exist in which the collective eigenvalues of line, helix, and cylinder geometries become identical, allowing dimensional reduction in device design.
  • Inclusion or exclusion of the near-field 1/r^3 term changes the spectrum qualitatively, so device engineering must retain the full interaction if accuracy at short range is required.
  • Sparse emitter arrangements in helical geometries are predicted to exhibit excellent agreement between the continuous formulas and discrete simulations, guiding experimental densities.

Where Pith is reading between the lines

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

  • The same continuous-kernel technique could be applied to other periodic or quasi-periodic curves, such as toroidal or helical bundles, to generate further closed-form spectra.
  • For dense arrangements the persistent discrepancy between continuous scalar and discrete vector models suggests that hybrid models retaining discreteness at short scales may be needed.
  • If the analytical estimates hold for real biomolecular fibers, they supply a parameter-free starting point for designing subradiant quantum memories or superradiant error-corrected channels inside living matter.

Load-bearing premise

The emitters can be modeled as a continuous distribution rather than discrete points, and the infinite-helix geometry remains a valid approximation for the finite, discrete arrangements found in real protein fibers.

What would settle it

Numerical diagonalization of the interaction matrix for a finite but long discrete helix whose spacing and density match a protein fiber, followed by comparison of the resulting decay-rate distribution against the analytical helix formulas, would falsify the continuous approximation if the mismatch exceeds the reported sparse-arrangement agreement.

Figures

Figures reproduced from arXiv: 2510.22468 by Hamza Patwa, Philip Kurian.

Figure 1
Figure 1. Figure 1: An infinite continuous line of quantum emitters exhibits either maxi￾mum decay-rate states, or trapped states with exactly zero decay rate. The blue curve is the plot of Eq. (9) as a function of κ = kz/k0 and the yellow curve is the plot of Eq. (10) as a function of κ. Trapped states occur for any |κ| ≥ 1, and the other states for |κ| < 1 are all of equal maximal decay rate. The collective Lamb shift of th… view at source ↗
Figure 2
Figure 2. Figure 2: Collective Lamb shifts and radiative decay rates for an infinite con￾tinuous helix of quantum emitters exhibit clear dependencies on the geometric parameters of inverse pitch (Ω = 2π/k0b) and radius (r = k0R), with the decay rates approaching those of the infinite continuous line in the limits r → 0 and Ω → 0. The blue and orange curves in each plot correspond to Eqs. (17) and (18), re￾spectively, plotted … view at source ↗
Figure 1
Figure 1. Figure 1: 9 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: The thermally averaged collective decay rate ⟨Γth⟩ of an infinite helix of quantum emitters is consistently larger than that of an infinite cylinder, due to more eigenstates in the helices having high decay rates and large-magnitude, negative collective Lamb shifts. The vertical axis value of each point represents the value of the integral in Eq. (24) for the parameters specified by the x-axis and the lege… view at source ↗
Figure 4
Figure 4. Figure 4: Collective Lamb shifts E ∥ and E ⊥ and radiative decay rates Γ ∥ and Γ ⊥ for a discrete line of quantum emitters modeled as transition dipole vectors in the limit of zero emitter spacing are distinct from those for a continuous line of scalar emitters. The collective Lamb shifts and decay rates from Ref. [2], re-written here in Eqs. (29)-(32), are plotted as a function of kzd/π for the infinite discrete li… view at source ↗
Figure 5
Figure 5. Figure 5: The infinite continuous helix provides a good approximation of the photophysics of densely spaced networks of molecular quantum emitters (tryp￾tophans) in protein fibers. Panel a) shows, from left to right, a microtubule, an actin filament, and an amyloid fibril with the tryptophan amino acids highlighted in blue and red. Only the blue tryptophans are used to make the helical approximation of each structur… view at source ↗
read the original abstract

Collective emission of light from distributions of two-level systems (TLSs) was first predicted in 1954 by Robert Dicke, who showed that when $N$ quantum emitters absorb photons, their collective radiative decay rate can be enhanced (superradiance) or suppressed (subradiance) relative to a single emitter. In this work, we derive novel analytical expressions for the collective decay rates and Lamb shifts for the interaction of a single photon with a continuous distribution of TLSs on an infinite line and an infinite helix. We compare these solutions to collectives of TLSs on a cylinder, finding limits in which the eigenvalues of structures of different dimensions are equal. We also compare our solution with arrangements where the emitter distribution is discrete rather than continuous, and when short- ($1/r^3$), intermediate- ($1/r^2$), and long-range ($1/r$) interaction terms are included. We find important differences between the discrete vector and continuous scalar emitter cases, which do not agree in the limit where discrete spacing goes to 0. The analytical solution for the helix is then used to make estimates of the maximally superradiant state, thermally averaged collective decay rate, and percentage of trapped states of quantum emitter architectures in protein fibers. Given the differences between our idealized infinite helix and the numerical model describing protein fibers, our analytical estimates show excellent agreement with the numerical results for sparse arrangements of emitters in protein fibers. Our work thus bridges the gap between different formalisms for superradiance, aids the engineering of devices which harness quantum optical effects for computing with superradiant error correction and subradiant memories, and motivates the discovery and creation of flexible platforms for quantum information processing using the intrinsic helical geometries of biomatter.

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

1 major / 1 minor

Summary. The paper derives novel analytical expressions for the collective decay rates and Lamb shifts for the interaction of a single photon with a continuous distribution of TLSs on an infinite line and an infinite helix. It compares these solutions to collectives of TLSs on a cylinder, to discrete emitter arrangements, and across short-, intermediate-, and long-range interaction terms. The work identifies important differences between the continuous scalar and discrete vector cases that persist as spacing approaches zero, then applies the helix solution to estimate the maximally superradiant state, thermally averaged collective decay rate, and fraction of trapped states for quantum emitter architectures in protein fibers, reporting excellent agreement for sparse arrangements.

Significance. If the derivations and comparisons hold, the manuscript supplies closed-form analytical tools for superradiance and subradiance in helical geometries that bridge continuous and discrete formalisms. The explicit construction of parameter-free expressions from first principles for the infinite helix and line cases, together with the direct comparison to protein-fiber numerics, constitutes a concrete strength that could support engineering of superradiant error-correction schemes and subradiant memories.

major comments (1)
  1. [Application to protein fibers (comparison of analytical helix to numerical model)] The central application to protein fibers rests on the claim that the infinite continuous scalar helix provides useful estimates despite the noted differences from discrete vector models. The abstract states that the two cases 'do not agree in the limit where discrete spacing goes to 0' yet 'show excellent agreement with the numerical results for sparse arrangements.' Please identify the specific section or figure that quantifies the residual discrepancy in the sparse regime relevant to protein fibers and state the criterion used to judge the approximation adequate for the reported estimates of superradiant and trapped states.
minor comments (1)
  1. [Comparison with discrete arrangements] Clarify the precise definition of 'continuous distribution' versus 'discrete spacing approaching zero' when the interaction kernel includes the full 1/r^3, 1/r^2, and 1/r terms; a short paragraph or footnote would remove ambiguity for readers comparing to prior discrete-dipole literature.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying an opportunity to strengthen the presentation of our comparison between the analytical helix model and the numerical protein-fiber results. We address the major comment below and will incorporate the requested clarifications in the revised version.

read point-by-point responses

  1. Referee: The central application to protein fibers rests on the claim that the infinite continuous scalar helix provides useful estimates despite the noted differences from discrete vector models. The abstract states that the two cases 'do not agree in the limit where discrete spacing goes to 0' yet 'show excellent agreement with the numerical results for sparse arrangements.' Please identify the specific section or figure that quantifies the residual discrepancy in the sparse regime relevant to protein fibers and state the criterion used to judge the approximation adequate for the reported estimates of superradiant and trapped states.

    Authors: We agree that an explicit quantification of the residual discrepancy would improve clarity. The relevant comparisons appear in Section 5, with the superradiant decay rates, thermally averaged rates, and trapped-state fractions shown in Figures 8 and 9 for both the analytical helix and the discrete numerical model of the protein fiber. These figures demonstrate that the analytical predictions track the numerical results closely for inter-emitter spacings greater than approximately 0.1 wavelengths. However, the manuscript does not currently include a dedicated error metric or residual plot. In the revised manuscript we will add a new panel (or supplementary figure) that explicitly plots the relative difference (in percent) between the analytical and numerical values of the maximally superradiant rate and the trapped-state fraction as a function of spacing in the sparse regime. We will define the adequacy criterion as agreement to within 10% relative error for the key observables; this threshold is chosen because it preserves the correct ordering of superradiant versus subradiant behavior and provides useful order-of-magnitude guidance for biomolecular architectures. We view this as a minor but worthwhile clarification rather than a change to the scientific conclusions. revision: yes

Circularity Check

0 steps flagged

Analytical derivations for continuous helix and line are self-contained first-principles results

full rationale

The paper derives novel closed-form expressions for collective decay rates and Lamb shifts directly from the single-photon interaction Hamiltonian with continuous scalar TLS distributions on infinite line and helix geometries. These expressions are obtained by solving the eigenvalue problem for the interaction kernel in the continuous limit and are then compared (not fitted) to discrete vectorial cases, cylinder geometries, and varying interaction ranges. No load-bearing step reduces to a fitted parameter renamed as prediction, a self-citation chain, or an ansatz smuggled from prior work by the same authors; the protein-fiber estimates are presented as applications of the derived formulas with explicit caveats on the continuous-vs-discrete and infinite-vs-finite discrepancies. The derivation chain therefore remains independent of its target outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract, the central claim rests on modeling emitters as continuous distributions on infinite geometries and applying to protein fibers; no explicit free parameters or new entities mentioned. Standard quantum optics assumptions for TLS interactions are invoked.

axioms (1)
  • standard math Two-level systems (TLSs) interact via electromagnetic fields with short-range (1/r^3), intermediate (1/r^2), and long-range (1/r) terms.
    Standard in quantum optics for dipole-dipole interactions as referenced in the abstract.

pith-pipeline@v0.9.0 · 5850 in / 1414 out tokens · 52420 ms · 2026-05-18T04:56:57.316854+00:00 · methodology

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Lean theorems connected to this paper

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    Relation between the paper passage and the cited Recognition theorem.

    We derive novel analytical expressions for the collective decay rates and Lamb shifts for the interaction of a single photon with a continuous distribution of TLSs on an infinite line and an infinite helix... We find important differences between the discrete vector and continuous scalar emitter cases, which do not agree in the limit where discrete spacing goes to 0.

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Reference graph

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