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Optimal Waveforms for Dipole Moment Estimation with Coherent States

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Pith's one-line read A half-cycle sine pulse maximizes the precision with which a coherent light pulse can reveal an atom's dipole moment, reaching a per-photon quantum Fisher information of 4.

desk verdict A careful, internally consistent ODE-based treatment of optimal coherent-state pulses for dipole estimation; the long-pulse theorem (QFI per photon ≤ 4 with sine/plane-wave optima) holds, with the lossless assumption and a numerically grounded claim about arbitrary-width complex pulses as the main caveats. read the letter →

arxiv 2509.09807 v1 pith:GORHVOMT submitted 2025-09-11 quant-ph

Karthik Chinni , Nicolás Quesada This is my paper · ORCID
classification quant-ph
keywords quantumFisherinformationdipolemomentestimationcoherentstateswaveguideelectrodynamicsopticalBlochequationspulseshapingsensingspontaneousemissionrate
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks which temporal shape of a weak coherent light pulse best reveals the dipole moment of a two-level atom, i.e. its spontaneous emission rate $\Gamma$, from the light the atom scatters. It derives a closed set of coupled ordinary differential equations whose solution gives the quantum Fisher information, the figure of merit that bounds estimation precision, directly and without finite-difference approximations. In the long-pulse limit the per-photon quantum Fisher information is bounded by 4, and the bound is reached by a pulse whose envelope is a half-cycle sine at frequency $\Gamma/2$, with a phase set by detuning; with periodic boundary conditions the same value comes from a plane wave at frequency $\delta \pm \Gamma/2$. This matters because it identifies both the ultimate precision limit and the explicit waveform that attains it for an experimentally relevant coherent-state probe.

What carries the argument

The central object is the generalized atomic density operator $\mu_{\theta_1,\theta_2}(t)$, evolved under the double-sided master equation, whose trace gives the fidelity between two copies of the system with different parameter values; the QFI is obtained as the second derivative of that fidelity with respect to the parameter shift. The paper's main technical step is to convert that second derivative into a closed system of coupled real ODEs (four ODEs for real pulses, eight in general), whose solutions give the QFI directly and which contain the optical Bloch equations as a subsystem. In the long-pulse limit the QFI becomes a bilinear form in the pulse shape, and diagonalizing the associated kernel in a harmonic basis yields eigenvalues $64\tilde n^2/(4\tilde n^2+1)^2$; the largest eigenvalue is 4 and occurs at $\tilde n=1/2$, i.e. frequency $\omega=\Gamma/2$. A three-term decomposition $F=F_p+F_z+F_x$ isolates the only potentially negative contribution $F_x$, which the optimal pulse cancels.

What would settle it

Run the paper's ODE system for a long, lossless pulse expanded in many harmonic coefficients and optimize the coefficients: if any mixture yields $F_\infty/\alpha^2 > 4$, the central bound is wrong. An experiment that estimates the dipole moment with a per-photon precision better than the corresponding 4-per-photon QFI would also refute it; introducing a small loss rate $\Gamma_\perp>0$ and checking whether the sine pulse remains optimal would test the load-bearing lossless assumption.

Watch

Extended reading notes

Core claim

The paper claims that for a two-level atom in a one-dimensional waveguide driven by a coherent-state pulse, with no losses and full access to the scattered light, the quantum Fisher information per input photon for estimating the atom's dipole moment saturates at 4 in the long-pulse limit. The waveform that reaches this value is $f(t)=\sqrt{2/T}\,e^{-i\delta t}\sin(\Gamma t/2)$ when the pulse is required to vanish at its endpoints; under periodic boundary conditions the same value is obtained by a plane wave $f(t)=\sqrt{\alpha/T}\,e^{i(\delta\pm\Gamma/2)t}$. For real pulses with zero detuning the optimal shape is $\sqrt{2/T}\sin(\Gamma t/2)$, and the per-photon QFI matches the single-photon QFI; for complex pulses a fourth-order term makes the coherent-state result differ from the single-photon result. The paper also establishes a set of coupled ODEs whose solution gives the QFI directly, decomposes the QFI into two positive contributions and one potentially negative contribution, and shows that in the short-pulse limit the rectangular pulse is optimal.

Load-bearing premise

The calculation assumes the atom emits only into the measured waveguide mode ($\Gamma_\perp=0$), so all scattered light is collected; if any emission is missed, the QFI bound and the optimal waveform could change.

Editorial extensions

If this is right

  • The per-photon QFI bound of 4 means that, in the lossless long-pulse regime, no coherent pulse shape can improve dipole-estimation precision beyond a fixed amount per photon; adding photons is the only linear scaling lever.
  • The explicit optimal waveform gives a recipe: shape a coherent pulse as a half-cycle sine at frequency $\Gamma/2$ (with a detuning phase) to saturate the bound.
  • Allowing complex pulse phases and detuning does not increase the achievable QFI per photon; real pulses already reach the same bound in the closed-boundary case.
  • The new ODE system replaces finite-difference QFI evaluation, so pulse optimization and multi-parameter estimation can be performed stably for arbitrary pulse shapes.
  • For very short pulses the rectangular pulse is optimal, and at large photon number the standard pulse families all approach the same per-photon QFI, so the precise shape matters less in that regime.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same ODE reduction could be applied to estimating other atomic parameters, such as detuning or coupling strength, where the optimal waveform may differ from the dipole-moment one.
  • Because $F_x$ is the only negative contribution, waveforms engineered to keep $x(t)$ small, for example by avoiding sign flips, may be near-optimal in finite-width settings even when the exact sine shape is not reachable.
  • Adding a small loss rate $\Gamma_\perp>0$ is the immediate stress test: the paper's bound assumes perfect collection of scattered light, and the optimal pulse may need to be re-optimized when emission into unobserved modes is included.
  • The equality with single-photon QFI for real pulses suggests the per-photon optimum is set by the atom's excitation amplitude rather than by photon statistics; probes with non-classical light could reveal whether the bound changes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The manuscript studies quantum-limited estimation of the dipole moment (spontaneous emission rate Γ) of a two-level atom coupled to a one-dimensional waveguide, when the probe is a coherent-state pulse and all emitted light is assumed accessible (Γ⊥=0). The main technical contribution is a set of coupled ordinary differential equations whose solution directly gives the global quantum Fisher information (QFI), avoiding finite-difference evaluation of the double-sided master equation. The authors analyze standard pulse shapes, derive analytic QFI expressions in the short- and long-pulse-width limits, and show that in the long-pulse limit the QFI per unit photon is at most 4. Under closed boundary conditions f(0)=f(T)=0 the optimizer is the harmonic sin(Γt/2) (with a detuning phase in the complex case), and under periodic boundary conditions it is the plane wave with frequency δ±Γ/2. Numerical optimization in harmonic and Hermite-Gaussian bases supports the value 4 in the long-pulse regime. The paper is careful to state that the lossless assumption is a boundary of the model and that a full treatment of losses is left for future work.

Significance. If the results hold, the paper provides a practically useful and analytically tractable prescription for waveform optimization in waveguide-QED spectroscopy, and it establishes a clean connection between coherent-state and single-photon pulse limits. The derivation is detailed and mostly self-contained: the ODE system is derived in Appendix B, the perturbation theory in Appendix F, and the kernel diagonalizations in Appendices H and I. The work uses no fitted constants, and the numerical results are supported by publicly available code. The main limitation is the explicit Γ⊥=0 assumption, which is acknowledged in the conclusion and bounds direct experimental applicability without undermining the idealized theorem. The paper is an honest mix of analytic limit theorems and numerical evidence, and the central long-pulse claim appears internally consistent.

minor comments (5)
  1. [Sec. IV D, Sec. V, App. H4] The stated optimal pulse is written inconsistently across the paper: Sec. IV D gives sqrt(2α²/T) sin(πt/2T), Sec. V gives sqrt(2/T)e^{-iδt} sin(Γt/2), and Appendix H4 gives sqrt(2α/T) sin(Γt/2). Only the form with sqrt(2α²/T) is consistent with the normalization ∫|f(t)|²dt=α² used in Eq. (13). The correct long-pulse optimal pulse should be sqrt(2α²/T) sin(Γt/2) for closed boundaries, and the text should state this normalization explicitly in every occurrence.
  2. [Sec. IV D and App. H4] For f(t)=sqrt(2α²/T) sin(Γt/2) to satisfy the closed boundary condition f(T)=0, the width must satisfy ΓT/2 = nπ for some integer n. The manuscript does not state this condition when presenting the optimal pulse for the long-pulse limit; please add this qualification so that the asymptotic statement is precise.
  3. [Sec. V, paragraph after Eq. (61)] The sentence 'In other words, the optimal pulse is given by ... with the maximum QFI, F∞/α²=4' overstates the result for arbitrary pulse width. The value 4 for complex pulses of arbitrary width is obtained from numerical optimization, not from the perturbation-theoretic proof that precedes it. Please qualify this statement as a numerical observation or restrict the definitive claim to the long-pulse limit.
  4. [App. E] Appendix E is introduced as showing that Fp/α² is positive and bounded above by 4, but the appendix only proves positivity; the upper bound is established later in Appendix H1. Please adjust the cross-reference so that readers are not misled about where each part of the claim is proven.
  5. [References] Reference [49] is cited as 'in preparation' and cannot be checked. Please replace it with a published version or remove it from the bibliography.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the optimal pulse and QFI bound 4 are obtained by diagonalizing a kernel derived from the external two-sided master equation, not by fitting or self-citation.

full rationale

The derivation chain is self-contained. The central ODEs (29)-(32) and (52)-(60) are obtained from the double-sided master equation (28), which is attributed to the external references [31,32]; the paper does not derive that equation from its own target result. The long-pulse QFI functional in Eq. (49) and its complex-pulse generalization in Eq. (62) are obtained by a systematic perturbation expansion in epsilon = alpha/sqrt(Gamma T), and the kernel matrices K1, K2, K3 in Appendices H and I are built explicitly from the functions p(t), q(t), P(t), Q(t). The value 4 is not imposed: it emerges as the maximum of the eigenvalues 64 ntilde^2/(4 ntilde^2+1)^2 over ntilde, and the corresponding eigenvector gives sin(Gamma t/2) or the plane wave exp(i(delta +/- Gamma/2)t). This is a mathematical maximization of a derived quadratic form, not a fit of the target QFI nor a renamed single-photon result. The single-photon comparisons are made after the fact: Appendix G derives the identity F_infinity_long = 16 integral (dq/dt)^2 and then recognizes it as the single-photon QFI expression from [18]; [18] is used as a benchmark, not as an input that fixes the coherent-state optimum. The numerical optimization section uses the analytically derived optimum as one of ten seeds, but nine seeds are random and the analytic derivation is independent, so the numerics serve as confirmation rather than as the source of the claim. The only self-citation of note is Ref. [38], a tutorial co-authored by one of the present authors; it supplies the standard atom-waveguide Hamiltonian and is not used to justify the optimal-pulse bound. There is no imported uniqueness theorem, no ansatz smuggled in via citation, and no fitted parameter renamed as a prediction. The explicit assumption Gamma_perp = 0, stated in Eq. (4) and Fig. 1, is a modeling boundary acknowledged by the authors and not a circular step. Overall, the central claim is supported by an internally derived optimization problem, and no load-bearing step reduces to its own inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data; alpha, T, delta, and Gamma are physical inputs and the estimated parameter. The derivation relies on standard quantum optics and the imported [32] QFI formalism. No new entities are postulated.

assumptions (6)
  • domain assumption The atom couples to a Markovian, dispersionless one-dimensional waveguide at a point, with the Hamiltonian of Eqs. (1)-(4).
    The interaction picture and the Lindblad master equation in Sec. II rest on this standard waveguide-QED simplification; it excludes non-Markovian and dispersive effects.
  • domain assumption No losses: Gamma_perp = 0, so all emitted light is in the accessible waveguide.
    Set in Sec. II after Eq. (4); it allows the bipartite QFI simplification via the two-sided master equation.
  • domain assumption The QFI equals -4 times the second derivative of the fidelity, as expressed through the generalized density operator of [32].
    Eqs. (25)-(28) import this external theorem without re-derivation; it is the backbone of the ODE derivation.
  • domain assumption The initial state is a coherent-state pulse in the waveguide and the atom in its ground state.
    Eq. (12) fixes the input; the Mollow transformation in Appendix A depends on this factorization.
  • domain assumption Regular perturbation theory to second order in epsilon = alpha / sqrt(Gamma T) is valid in the long-pulse limit.
    Sec. IV D and Appendix F assume epsilon << 1; the optimal-pulse diagonalization is derived in this regime.
  • domain assumption The atom relaxes completely to the ground state in the long-time limit, so the global and emitted-field QFI coincide.
    Used at the start of Sec. IV; it follows from the Lindblad equation with no driving after the pulse, but is an assumption about the asymptotic state.

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Cite this review

Pith. "Pith review of Optimal Waveforms for Dipole Moment Estimation with Coherent States." pith.science (2026). https://pith.science/paper/GORHVOMT

@misc{pith2026250909807,
  author       = {Pith},
  title        = {Pith review of: Optimal Waveforms for Dipole Moment Estimation with Coherent States},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GORHVOMT}},
  note         = {Machine review of arXiv:2509.09807}
}
read the original abstract

We investigate quantum sensing for spectroscopy in a system consisting of a two-level atom coupled to a continuum of modes. We focus on optimizing the pulse shape of a coherent state to maximize the quantum Fisher information (QFI) of the emitted light with the aim of estimating the atom's dipole moment, which is proportional to its spontaneous emission rate. To achieve this, we derive a set of coupled differential equations, which include the standard optical Bloch equations as a subset and whose solution directly yields the QFI of the emitted light without resorting to finite-difference methods. Furthermore, we analyze the factors that govern its optimization, provide analytic solutions in both the long and the short pulse width limits, and examine the role of the average photon number of the pulses. We then show that under the closed (periodic) boundary conditions, the harmonic (plane-wave) with frequency equal to half the spontaneous emission rate and a phase determined by detuning are optimal in the long pulse width limit.

Figures

Figures reproduced from arXiv: 2509.09807 by the authors.

Figure 1
Figure 1. Our setup consists of a two-level atom coupled [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. (a) The QFI per unit photon, F ∞/α2 = (F ∞p + F ∞z + F ∞x )/α2 , of standard pulses plotted as a function of the variance of the pulse Tσ for α 2 = 1. See Table (I) for more details about the pulses. All these pulses show similar behavior with Tσ: the QFI initially increases with the pulse width, reaches a maximum, and then decreases toward zero as the width becomes larger. (b, c) Both F ∞p /α2 and F ∞z /α2 are alwa… view at source ↗
Figure 3
Figure 3. The QFI per unit photon of α 2 = 10−2 for the standard pulses is plotted in (a) while the contributions F ∞p /α2 , F ∞z /α2 and F ∞x /α2 are plotted in (b), (c) and (d) respectively. These quantities are all defined below Eq. (33), and satisfy F ∞ = F ∞p + F ∞z + F ∞x , with F ∞x being the only term that can have negative contribution. (e-h): Same as (a-d) for α 2 = 102 .(i-l): Same as (a-d) for α 2 = 104 . The QFI … view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: (a) The first three harmonics ξn = sin(nπt/T) are shown here. (b) The QFI of the numerically optimized pulse, expressed in the basis of harmonics, is shown for ΓT = 700 (stars), ΓT = 2000 (triangles) and ΓT = 5000 (dots) for nmax = 1000, which is the maximum number of …
Figure 5
Figure 5. Figure 5: (a) The first three Hermite-Gaussian functions are shown here. (b) The QFI of the numerically optimized pulse, expressed in the Hermite–Gaussian basis, is shown for nmax = 250 (stars) and nmax = 500 (dots) where nmax is the maximum number of basis functions considered.…
Figure 6
Figure 6. Figure 6: (a) Maximum eigenvalue as a function of the width of the pulse ΓT = rπ for nmax = 5, nmax = 50 and nmax = 2000 where nmax is the maximum number of basis functions considered. As seen, the maximum eigenvalue is 4 for all r. (b-d) The population of the eigenvector corres…

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    For code, see https://github.com/polyquantique/ OptimalWaveformsForDipoleEstimation

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    D. J. Griffiths and D. F. Schroeter,Introduction to quan- tum mechanics(Cambridge university press, 2018). Appendix A: Mollow transformation In this section, we show that the state of the system at an arbitrary time can be written as the expression given in Eq.(14) if the init...

    2018

  46. [54]

    These functions go to zero at the boundaries and form an orthonormal basis for square integrable functions that are defined betweent= 0 andt=T

    Harmonics The harmonics are defined as [53] ξn(t) = r 2 T sin nπt T ,(D1) where n ={1, 2,...} . These functions go to zero at the boundaries and form an orthonormal basis for square integrable functions that are defined betweent= 0 andt=T. The orthonormality condition is given...

  47. [55]

    Any real or complex function can then be expressed with the boundary conditionf(0) =f(T) = 0 as f(t) = r 2 T ∞X n=1 cn sin nπt T ,(D5) where cn is complex or real depending on whether the function f(t) is real or complex, and the coefficients are normalized to the average phot...

  48. [56]

    Hermite Gaussian basis The Hermite-Gaussian functions are defined as [53] ψn(t) := 1p 2nn!√π Hn(t)e−t2/2,(D7) where Hn(t) = (−1)net2 dn dtne−t2 .(D8) We will rescale these functions so that they take account into account a parameter T , which is proportional to the variance of...

    2000

  49. [57]

    Evaluation ofF 1 Note that F1 is identical to Fp, which is one of the terms that appears in the QFI expression without any approximations as shown in Eq. (33). This term can be rewritten in terms of harmonic functions as F1(t→∞) =F ∞ 1 = X m,n cm " 2 Γ Z T 0 dτ Z τ 0 dτ′e− Γ(τ...

  50. [58]

    Evaluation ofF 2 Note thatFz in the expression for the QFI becomes F2 in the long pulse width limit. Setting the width of the pulse as ΓT=rπ,F 2 in the long-time limit is given by F2(t→∞) =F ∞ 2 ≡ 1 Γ X m,n cm Z ∞ 0 dτ ϕm(τ)ϕn(τ) cn (H17) ≡ X m,n cm [K2(r)]m,ncn.(H18) where [K...

  51. [59]

    Evaluation ofF 3 In the long-time limit with ΓT=rπ, the final termF 3 is given by F3(t→∞)≡F ∞ 3 = 4 Z ∞ 0 dτ f(τ)q(τ) (H22) = X m,n cm 4 Z T 0 dτξm(τ)ηn(τ) ! cn (H23) ≡ X m,n cm [K3(r)]m,ncn,(H24) 24 where [K3(r)]m,n =    64mnr2 h r((4n2+r2)2−(−1)m+n(4m2+r2)2)+(−1)me−πr 2 (...

  52. [60]

    Maximum eigenvalue In summary, we have Γ2F∞ long α2 = Γ2F∞ 1 α2 + Γ2F∞ 2 α2 + Γ2F∞ 3 α2 (H32) = 1 α2 X m,n cm 8 4˜n2−1 δm,n (4˜n2 + 1)2 + 8δm,n 4˜n2 + 1 ! cn = X m,n ˜cm 64˜n2δm,n (4˜n2 + 1)2 ˜cn.(H33) Note that the coefficients {˜cm} are normalized, and the eigenvalues are th...

  53. [61]

    Perturbation theory for complex pulses The equations of motion with complex pulses and nonzero detuning given in Eqs. (52)-(60) can be written in a more compact form, r′ 1(t) =−kr 1(t) + 2 √ Γf∗(t)z 1(t),(I1) z′ 1(t) =−Γ (1 +z 1(t))−2 √ Γℜ f(t)r1(t) ,(I2) w′ 1(t) = 1 2 √ Γ ℑ f...

  54. [62]

    In this subsection, the integrals are extended from −∞ to∞ under the implicit assumption that f(t) = 0 for t≤ 0, thereby facilitating the use of Fourier transforms

    QFI of complex single-photon pulse with nonzero detuning The QFI for a single-photon pulse with complex pulse shapef(t) in the long-time limit is provided in [19], F∞ single = 1 Γ Z ∞ −∞ dω ˜f(ω) 2 |g(ω) + 2Γ∂Γg(ω)|2− 1√ Γ Z ∞ −∞ dω ˜f(ω) 2 1− √ Γg∗(ω) (g(ω) + 2Γ∂Γg(ω)) 2 ,(I4...

  55. [63]

    Diagonalizing QFI up to second order with the closed boundary conditions The QFI in the long pulse width limit (Eq. (I39)) is given by Flong(t) = 2 Γ3/2 Z t 0 dτℜ eiδτf(τ)P(τ) + 1 Γ Z t 0 dτ|P(τ)| 2 + 4 Z t 0 dτℜ eiδτf(τ)Q(τ)) (I62) ≡F 1(t) +F 2(t) +F 3(t) (I63) where P(t) = √...

  56. [64]

    16n3π+nr(4 +πr) (16n2 +r 2)2 − m 16m2π+r(4 +πr) (16m2 +r 2)2 # (I105) + 32r3 πΓ2(m−n)

    Diagonalizing QFI up to second order with the periodic boundary conditions In the plane-wave basis, we have f(t) = P ncnξn(t) = p 1/TP ncneiωnt where ωn = 2 nπ/T and n = {...,− 2,− 1, 0, 1, 2,...} due to periodic boundary conditions, f(0) = f(T ). We first assume that the detu...

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Reviewed August 15, 2026 · model on record in the stance chip above.