arxiv: 2605.05434 · v1 · submitted 2026-05-06 · 🪐 quant-ph

Recognition: unknown

Non-Markovian delay-assisted sensing with waveguide-coupled quantum emitters

Prajit Dhara , Isack Padilla , Saikat Guha , Annyun Das , Kanu Sinha

Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:16 UTC · model grok-4.3

classification 🪐 quant-ph

keywords waveguide quantum electrodynamicsnon-Markovian dynamicsquantum sensingquantum Fisher informationfield gradient sensingtime-delayed feedbackatom-photon bound statesdistributed quantum sensing

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The pith

Distant separations between two waveguide-coupled quantum emitters turn time delay into a resource that increases the quantum Fisher information for estimating external field gradients.

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

The paper establishes that in a minimal two-emitter waveguide system placed far apart under an external field, the non-Markovian dynamics arising from propagation delays can improve precision in sensing the field's gradient. The gradient appears as a detuning between the emitters, and the delays permit atom-photon quasi-bound states to form, letting the field interact with the emitters over longer times and through multiple modes. This converts what is usually treated as a complication into an advantage for extracting more information about relative detunings. A reader would care because it points to a way of using simple, scalable waveguide architectures for distributed quantum sensing without needing complex control.

Core claim

In a minimal setup of two waveguide-coupled quantum emitters separated by long distances and subject to an external field, time-delayed feedback enhances the quantum Fisher information for estimating the detuning parameter, and thereby the field gradient. The enhancement is attributed to the formation of atom-photon quasi-bound states that enable longer interaction times and to the mediation of interactions via multiple spectral modes of the field.

What carries the argument

Atom-photon quasi-bound states created by non-Markovian time-delayed feedback, which prolong field interaction and allow multimode mediation between the emitters.

If this is right

The quantum Fisher information for detuning estimation rises when non-Markovian delays are present compared with the Markovian limit.
Multimode field mediation supplies an additional channel that further improves sensing performance.
Large separations can be engineered as a controllable resource rather than a source of decoherence.
Minimal waveguide setups become viable for distributed sensing of spatial field variations.

Where Pith is reading between the lines

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

The same delay mechanism could be tested in chains of three or more emitters to map higher-order field derivatives.
Tunable delay lines in integrated photonic circuits would let experimenters sweep separation to find the separation that maximizes information for a given gradient strength.
The approach may connect to existing waveguide QED experiments where non-Markovian revivals are already observed, providing a near-term testbed.
If the quasi-bound states persist under realistic loss, the scheme could be combined with error-correction protocols for longer sensing times.

Load-bearing premise

That large interatomic separations produce non-Markovian dynamics in which atom-photon quasi-bound states form and meaningfully extend the time the field spends interacting with the emitters.

What would settle it

A direct comparison of quantum Fisher information values extracted from the two-emitter system at large separations versus small separations, under identical field strengths, to check whether the information increases with increasing delay.

Figures

Figures reproduced from arXiv: 2605.05434 by Annyun Das, Isack Padilla, Kanu Sinha, Prajit Dhara, Saikat Guha.

**Figure 1.** Figure 1: FIG. 1. (a) Two resonant two-level atoms coupled to a view at source ↗

**Figure 2.** Figure 2: FIG. 2. Emitter and field dynamics in the presence of de view at source ↗

**Figure 3.** Figure 3: shows the emitted field spectrum G (˜ω) as a function of frequency ˜ω and the interatomic separation η, for three separate values of the detuning δ. We highlight the salient features of the field spectrum as follows view at source ↗

**Figure 4.** Figure 4: FIG. 4. Quantum Fisher information ( view at source ↗

**Figure 5.** Figure 5: FIG. 5. Quantum Fisher information view at source ↗

**Figure 6.** Figure 6: FIG. 6. Atomic spectral response functions view at source ↗

**Figure 7.** Figure 7: FIG. 7. Plot of QFI view at source ↗

read the original abstract

We show that in a minimal setup of two waveguide-coupled quantum emitters, separated by long distances and subject to an external field, time-delayed feedback can be a resource for sensing field gradients. While the field gradient induces a detuning between the emitters; the large interatomic separations render the system dynamics non-Markovian. We show that the quantum Fisher information (QFI) for estimating the detuning parameter, and thereby the field gradient, is enhanced in the presence of non-Markovian delay. Such an enhancement can be attributed to the formation of atom-photon quasi-bound states that enable the field to interact with the emitters for longer times, thereby gaining more information about their relative detunings. Additionally, in the presence of delay, the interaction between the emitters is mediated via multiple spectral modes of the field, further enhancing the sensing capabilities of the system. Our results establish non-Markovian time-delayed feedback and multimode reservoirs as a resource for distributed quantum sensing with waveguide-coupled 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.

Desk Editor's Note private letter to a colleague

The paper shows QFI gains from non-Markovian delay in a two-emitter waveguide sensor, but the link to quasi-bound states needs tighter isolation from other delay effects.

read the letter

The core result is that large separations between two waveguide-coupled emitters make the dynamics non-Markovian and raise the quantum Fisher information for estimating the detuning from an external field gradient. The authors tie the improvement to atom-photon quasi-bound states that extend interaction time plus multimode mediation of the coupling. The setup is deliberately minimal, which helps keep the focus on the delay resource rather than extra complexity. That framing of non-Markovianity as useful for distributed sensing is the clearest new angle here, and the multimode point follows naturally once retardation is included. The calculations appear to rest on standard waveguide QED techniques, and the claim that delay opens extra spectral channels for information is plausible on its face. The soft spot is exactly the one flagged in the stress-test note. The enhancement is attributed to the quasi-bound states, yet the paper does not appear to supply a controlled comparison that keeps delay while suppressing bound-state formation, nor does it show QFI scaling directly with the bound-state lifetime. Recovering the Markovian limit inside the same model would also make the contrast sharper. Without those steps the causal claim stays a bit loose, even if the overall numerics or analytics look reasonable. This is for readers already working in waveguide QED or quantum metrology who want to explore time-delay resources. Someone building sensors or studying non-Markovian effects would get concrete ideas to test or extend. The work is coherent enough on its own terms to deserve referee time, even if revisions are likely needed on the mechanism.

Referee Report

2 major / 2 minor

Summary. The manuscript studies a minimal model of two quantum emitters coupled to a one-dimensional waveguide and separated by large distances, subject to an external field gradient that induces a detuning. It claims that the resulting non-Markovian time-delayed feedback enhances the quantum Fisher information (QFI) for estimating this detuning (and hence the gradient), with the enhancement attributed to the formation of atom-photon quasi-bound states that prolong the interaction time and to multimode mediation of the inter-emitter coupling.

Significance. If the central claim is substantiated with explicit derivations, this work would establish non-Markovian delay as a concrete resource for distributed quantum sensing in waveguide QED, complementing existing Markovian approaches and highlighting multimode reservoirs. The use of QFI as the figure of merit is appropriate and the minimal two-emitter setup is a strength for isolating the mechanism.

major comments (2)

[Abstract and §3] Abstract and §3 (Results): The attribution of QFI enhancement specifically to atom-photon quasi-bound states is load-bearing but not yet secured. No pole analysis of the resolvent, long-time asymptotics of the excitation amplitude, or controlled comparison (delay present but bound-state formation suppressed) is referenced to isolate this effect from retarded propagation or spectral-density changes alone.
[§2 and §4] §2 (Model) and §4 (Discussion): The Markovian limit must be recovered within the same framework (zero delay or instantaneous feedback) to demonstrate that the reported QFI increase is due to non-Markovianity rather than multimode effects that could persist in the Markovian regime. Without this benchmark, the causal role of delay remains unconfirmed.

minor comments (2)

[Figure captions and §3] Figure captions and §3: Clarify whether the plotted QFI curves include the Markovian reference case or only the delayed case; explicit overlay would strengthen the comparison.
[§2] Notation in §2: The definition of the detuning parameter Δ and its relation to the field gradient should be stated with an explicit equation to avoid ambiguity in the sensing task.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the presentation of our results on non-Markovian delay-assisted sensing. We address each major comment below.

read point-by-point responses

Referee: [Abstract and §3] Abstract and §3 (Results): The attribution of QFI enhancement specifically to atom-photon quasi-bound states is load-bearing but not yet secured. No pole analysis of the resolvent, long-time asymptotics of the excitation amplitude, or controlled comparison (delay present but bound-state formation suppressed) is referenced to isolate this effect from retarded propagation or spectral-density changes alone.

Authors: We appreciate the referee pointing out the need for more explicit evidence isolating the quasi-bound-state mechanism. Our §3 analysis correlates the QFI enhancement with the delay-induced prolongation of interaction times and multimode coupling, supported by numerical dynamics of the excitation amplitudes. To strengthen this, the revised manuscript will incorporate a pole analysis of the resolvent to identify the quasi-bound states, the corresponding long-time asymptotics of the amplitudes, and a controlled comparison (with delay retained but bound-state formation suppressed via parameter tuning) to distinguish from pure retardation or spectral-density effects alone. revision: yes
Referee: [§2 and §4] §2 (Model) and §4 (Discussion): The Markovian limit must be recovered within the same framework (zero delay or instantaneous feedback) to demonstrate that the reported QFI increase is due to non-Markovianity rather than multimode effects that could persist in the Markovian regime. Without this benchmark, the causal role of delay remains unconfirmed.

Authors: We agree that an explicit benchmark against the Markovian limit within the same model is necessary to confirm the role of non-Markovian delay. The original manuscript discusses the small-delay limit, but the revised version will add a dedicated recovery of the Markovian case (zero delay or instantaneous feedback) in §2, together with direct QFI comparisons in §4. These will show that the reported enhancement and multimode mediation require the finite delay and do not appear in the Markovian regime. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains independent of its target result

full rationale

The paper models two waveguide-coupled emitters with time-delayed feedback using standard non-Markovian quantum optics (delay differential equations or resolvent techniques for the field-mediated interaction). The QFI for the detuning is then computed directly from the resulting time-evolved state or density matrix under the gradient-induced detuning. No step defines the enhancement in terms of itself, renames a fitted parameter as a prediction, or relies on a self-citation chain whose uniqueness theorem is invoked to force the outcome. The attribution to quasi-bound states follows from the pole structure of the delay kernel, which is derived from the waveguide dispersion relation rather than from the sensing figure of merit. The Markovian limit is recovered by taking the delay to zero within the same formalism, providing an internal consistency check that does not presuppose the claimed enhancement. Consequently the central claim does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

Ledger extracted from abstract claims only; full paper likely contains explicit model parameters and derivations not visible here.

free parameters (1)

inter-emitter separation distance
Large separation is invoked to induce non-Markovian dynamics but no specific value or fitting procedure is given in the abstract.

axioms (1)

domain assumption Waveguide-coupled emitters separated by long distances exhibit non-Markovian dynamics due to time-delayed feedback.
Directly stated in the abstract as the basis for the setup.

invented entities (1)

atom-photon quasi-bound states no independent evidence
purpose: Enable longer interaction times between the field and emitters to gain more information about detuning.
Introduced in the abstract to explain the QFI enhancement mechanism.

pith-pipeline@v0.9.0 · 5482 in / 1340 out tokens · 50382 ms · 2026-05-08T16:16:39.039653+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

81 extracted references · 5 canonical work pages · 1 internal anchor

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Atomic dynamics: Series solution We solve the coupled dynamics of the atoms by expanding the denominators in the Laplace coefficients in terms of a series, which physically corresponds to an expansion in terms of the number of photon wavepacket oscillations. We 11 rewrite the Laplace coefficientC 2(s) in Eq.(A4b) as γC2(s) = −βeiϕ2 e−η˜s/2 (˜s+ 1/2)(˜s+iδ...
[2]

(A4a)–(A4b) using Cauchy’s residue theorem

Atomic dynamics: Multiexponential decay As an alternate approach for solving the atomic dynamics, one can invert Eq. (A4a)–(A4b) using Cauchy’s residue theorem. In order to apply Cauchy’s residue theorem, we need to numerically evaluate the poles of Eq. (A4a) and (A4b), i.e., determine the roots of the denominators, f1(˜s;δ) = (˜s+ 1/2)(˜s−iδ+ 1/2)−β2(eiϕ...
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(14)), we get: G(¯ω) = γ π |F1(¯ω)|2 +|F 2(¯ω)|2 + 2 cos ¯ωη+δη 2 Re [F∗ 1 (¯ω)F2 (¯ω)] .(B9)

Field spectrum The emitted field spectrumG(˜ω), in the steady state, is given by G(˜ω)≡ |ca(˜ω,∞)|2 +|c b(˜ω,∞)|2 = γ π |F1(˜ω)|2 +|F 2(˜ω)|2 + 2 cos (˜ωη) Re [F∗ 1 (˜ω)F2 (˜ω)] .(B6) Shifting the frequency axis to be centered at ˜ω0, the explicit expression for the spectrum becomes: 14 G(¯ω) = γ π h |F1(¯ω)|2 +|F 2(¯ω)|2 + 2 cos ((¯ω+ ˜ω0)η) Re [F∗ 1 (¯ω...
[4]

Spectrum peaks in presence of detuning Equating the approximate peaks of the spectrum obtained via Eq. (24) to the resonance frequency of emitter 1, we get the following transcendental relation between the detuningδand the atomic separationη n ˜ω± nδ = ˜ω1 (B10) =⇒˜ω0 − δ 2 + 1 ηn Im h Wn ± ηn 2 eηn/2 i = ˜ω0 + δ 2 (B11) =⇒Im h Wn ± ηn 2 eηn/2 i =δη n.(B1...
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This can be seen qualitatively from Fig

Large detunings,δ >1: Forδ >1, such that emitter detunings are larger than their respective linewidths, a large number (n) of branches of the Lambert-W functionW n(z± n ) are required to describe the spectrum of the field close to the emitter resonances. This can be seen qualitatively from Fig. 3. Thus forδ≳1, the heuristic relationδη n ∼nπ, implies thatn...
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Thus forδ≪1, the heuristic relationδη n ∼nπ, implies thatnπ≪η n

Small detunings,δ≪1: For small emitter detunings, the corresponding values ofnfor the Lambert-W function branches needed to describe the spectrum of the field near emitter resonances are small. Thus forδ≪1, the heuristic relationδη n ∼nπ, implies thatnπ≪η n. In this regime, tan −1 2nπ ηn/2+ln(ηn/2) ≈0, such that Eqs. (B16) and (B18) yield: Im Wn z+ n ≈2nπ...
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