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REVIEW 2 major objections 3 minor 53 references

Detection-resolution limits of large-momentum-transfer atom gravimetry

T0 review · 2 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Detector resolution, not photon momentum, sets the ultimate sensitivity of fringe-readout atom gravimeters.

desk verdict Solid Fisher-information analysis of mirrorless LMT, but the claimed resolution floor is an artifact of fixing the mirrorless endpoint—asymmetry plus large n makes the fringe CFI unbounded within the same model. read the letter →

arxiv 2607.28749 v1 pith:TNE6EGVW submitted 2026-07-30 quant-ph

Asad Ali , Saif Al-Kuwari This is my paper · ORCID
classification quant-ph
keywords atominterferometrylargemomentumtransfergravimetryFisherinformationdetectorresolutionmomentum-resolvedreadoutmirrorlessinterferometerquantumsensing
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

Large momentum transfer (LMT) multiplies the gravitational phase of a light-pulse atom interferometer by n, and removing the mirror pulse in favor of momentum-resolved readout can quadruple the phase-carrying quantum Fisher information — the standard measure of how much a quantum state can tell about a parameter. This paper shows the two gains cannot be combined indefinitely: the fringes that carry the phase shrink to a period 1/n in momentum space while the detector's blur stays fixed, so the contrast decays and the extractable information is capped. In a solvable model with a Gaussian atomic source and Gaussian detection blur, the classical Fisher information is maximized at the momentum transfer n* ≈ 0.93 m/(σ_p k0 T), giving a resolution-limited sensitivity floor Δg ≈ 4.4 σ_p/(mT√N) that is independent of photon momentum. The resulting design rule is concrete: a mirrorless sequence beats a conventional equal-momentum-transfer sequence only when σ_p < 0.91 m/(n k0 T), which for rubidium sensors confines the mirrorless advantage to short-baseline instruments.

What carries the argument

The analytical engine is the exact momentum-space composition of the π/2–T1–π–T2–π/2 pulse family (mirror asymmetry s=(T1−T2)/2). Each output port is a Gaussian envelope times a cosine fringe of wave number κ=Δz/ℏ; a Gaussian convolution lemma gives the blurred distribution: contrast multiplied by C_e = exp(−κ²σ_p²δp²/2Σ²), wave number stretched to κ_e = κδp²/Σ², and phase shifted. Two-port cross terms cancel exactly, leaving F_C = (∂_g φ̃_g)² R(C_e²) + m²T²/Σ² with R(C²)=1−√(1−C²). Maximizing the fringe term over n reduces to a fixed-point equation whose unique positive solution a*=0.864 gives h*=0.2068, hence n* and the ceiling. This converts the trade-off into closed-form design rules.

What would settle it

Measure the output momentum distribution of a mirrorless atom gravimeter at fixed interrogation time and detector resolution, and estimate the classical Fisher information as a function of n from repeated estimates of g. If the information does not peak near n* ≈ 0.93 m/(σ_p k0 T) and then fall off with the predicted contrast factor exp(−σ_p² n² k₀² T²/2m²), the claimed ceiling is wrong. A simpler check: if a velocity-selective diffraction pulse with acceptance narrower than the source width δp changes the sensitivity by more than the model's envelope term allows, the Gaussian-envelope assumpt

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Extended reading notes

Core claim

The central claim: fringe-based momentum readout of a mirrorless atom gravimeter is a competition between the phase gain of large momentum transfer and the contrast loss from detector blur, and this competition has a sharp optimum. For Gaussian source and detector, each port's output is a Gaussian envelope times a cosine fringe of wave number growing with n; blur reduces contrast by exp(−σ_p²Δz²/2ℏ²). The Fisher information separates into envelope and fringe terms; maximizing the fringe term gives n* ≈ 0.93 m/(σ_p k0 T), sensitivity floor Δg ≈ 4.4 σ_p/(mT√N), independent of photon momentum, and crossover σ_p < 0.91 m/(n k0 T) for the mirrorless scheme to beat a conventional equal-n sequence.

Load-bearing premise

The calculation rests on the idealization that each beam splitter instantaneously transfers exactly n photon momenta with no velocity dependence and that the atomic source and detection blur are Gaussian; if real high-order diffraction pulses filter the momentum distribution or leave the internal state unlabeled, the fringe contrast and the envelope width change, and the quoted numerical thresholds will shift.

Editorial extensions

If this is right

  • A fixed detector resolution makes the extractable Fisher information grow with momentum transfer only up to n* ≈ 0.93 m/(σ_p k0 T); beyond that, extra recoils compress the fringes faster than they enlarge the phase, and the extractable information falls off.
  • The best per-shot sensitivity with fringe readout is Δg ≈ 4.4 σ_p/(mT√N); improving the detector's momentum resolution improves this directly, while adding photon momentum beyond the optimum does not.
  • A mirrorless sequence beats a conventional equal-n sequence only when σ_p < 0.91 m/(n k0 T). When that fails, partial mirror asymmetry (s between 0 and T/2) recovers part of the enhancement, with the optimized separation kept below the detector's resolving length.
  • Population-based readout, because its signal is a global phase rather than a momentum-space fringe, is insensitive to detector blur and grows as n²; at large n it favors the conventional sequence over the mirrorless one.
  • For rubidium, the long-baseline case (T=260 ms) requires σ_p below about 2×10⁻⁵ ℏk0 even at n=1, beyond current capability; the short-baseline case (T=10 ms) yields n* ≈ 5 and a mirrorless advantage up to n≈4.

Reading between the lines

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

  • The same fringe-compression mechanism should appear in any readout that extracts phase from momentum-space or spatial fringes, such as phase-shear and point-source imaging; the numerical constants will shift, but the qualitative picture of a resolution-limited optimum in momentum transfer is likely general.
  • A practical consequence the authors leave implicit: for compact gravimeters, improving detector resolution is as valuable as increasing momentum transfer, because the ceiling scales linearly in σ_p while the n² gain saturates at the optimum.
  • Because the controlling parameter is the dimensionless combination σ_p k0 T/m, the same design curves should transfer to other atomic species with equal values of this parameter — a testable scaling beyond the rubidium numbers in the paper.
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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

2 major / 3 minor

Summary. The paper studies a solvable model of a light-pulse atom gravimeter with instantaneous lossless n-photon beam splitters, a Gaussian source, and Gaussian detection blur, continuously interpolating between Kasevich–Chu and mirrorless geometries. It derives closed-form expressions for the momentum-space output distributions and the classical Fisher information for momentum-resolved readout, including a fringe-phase-averaging approximation with exponentially small error. The central claims are: (i) mirrorless operation delivers a fourfold QFI enhancement over KC but only outperforms an equal-n KC device when σ_p < 0.91 m/(n k0 T); (ii) fringe-based mirrorless readout has an optimal momentum transfer n* ≈ 0.93 m/(σ_p k0 T) and a resolution-limited sensitivity floor Δg_min ≈ 4.4 σ_p/(mT√N); (iii) partial mirror asymmetry recovers part of the gain when this criterion is violated. The derivations are validated against direct numerical simulation of the pulse sequence at the 10^-8 level.

Significance. The paper is a technically thorough contribution with several strengths: closed-form propagator-level solutions, two independent derivations of the QFI, exact (up to a well-controlled average) CFI expressions, and numerical validation at high precision. The parameter-free constants (e.g., h* = 0.20683) are derived, not fitted, and the paper is honest about the idealized nature of the model. If the central 'resolution ceiling' claim were correct, the design criteria would be directly useful for LMT gravimeter architecture. However, the ceiling as stated is not a bound on the interpolating family considered in the same paper, and this undermines the abstract-level claim. The paper's analysis of the mirrorless endpoint is sound, but the generalization to 'fringe-based readout' without qualification is not.

major comments (2)
  1. [Secs. IV and V, Eqs. (15), (23)] The claimed 'resolution-limited sensitivity floor' (Eqs. 19–20) is not a bound on the model's fringe-based readout. For the asymmetric-timing family, take u→0 and n→∞ while keeping x ≡ β u² fixed. From Eq. (23), F_fringe/(n²k0²Tπ⁴) = [2-(1-u)²]² R(e^{-x}) → R(e^{-x}) as u→0, while n² ∝ β = x/u² diverges as 1/u². Thus F_fringe ∝ x/u² → ∞ for any fixed x>0, and the exact CFI (Eq. 14) shares this divergence because ∂_g φ̃_g ≈ n k0 Tπ² → ∞ while the contrast factor remains constant. The ceiling h*m²T²/(4σ_p²) is an artifact of fixing the mirrorless endpoint u=1; the full model has no finite supremum. This directly contradicts the abstract's 'resolution-limited sensitivity floor' and the statement that Eq. (19) is a ceiling on fringe-based readout. The authors must either restrict the ceiling claim explicitly to the mirrorless configuration (and revise the abstract), or analyze the joint (n,u
  2. [Abstract and Sec. IV] The abstract states 'Fringe-based readout exhibits an optimal momentum transfer n* ... and a resolution-limited sensitivity floor Δg≃4.4σ_p/(mT√N), independent of photon momentum.' This is misleading because n* and the floor are derived only for the mirrorless case (s = Tπ). For the interpolating family, the infimum of Δg over s and n is zero within the model. The paper should explicitly qualify these results as applying to the mirrorless configuration, and should discuss whether any physical constraint (e.g., finite pulse duration, velocity acceptance, or an upper bound on n) restores a finite optimum. This is not merely cosmetic: the practical conclusions about when to use conventional LMT versus mirrorless operation depend on whether the asymmetric family can circumvent the resolution penalty.
minor comments (3)
  1. [Appendix D, Eq. (D12)] The relative error bound is stated as K(C_e) e^{-2κ_e²Σ²}, but K(C_e) is not defined or bounded. A short derivation or an explicit expression for K in terms of Fourier coefficients would make the claim more quantitative.
  2. [Fig. 4] The figure fixes n=5 and shows the normalized CFI across u. This is useful for fixed-n comparison, but the text should caution that the joint optimization over (n,u) is not shown and, as discussed, leads to unbounded CFI in the idealized model.
  3. [Sec. VI, first paragraph] The sentence 'The resolution floor, Eq. (20), is Δg√N≈5×10^{-4} m s^{-2} for these numbers' uses the mirrorless floor without noting that the asymmetric family could, in principle, give better sensitivity in the same idealized model. The practical estimates should state this caveat explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained, all constants come from the stated model, and no self-citation is load-bearing.

full rationale

The paper's derivation chain is explicit and non-circular. Section II and Appendices A/B solve the pulsed model in closed form and derive the QFI (Eq. 5) by two independent routes (generator conjugation and branch-amplitude moments), rederiving rather than merely importing the external result of Ref. [25]. Section III and Appendices C/D obtain the blurred CFI (Eq. 14) by closed-form Gaussian convolution; the only approximation, the fringe-phase average, has an exponentially small error bound and is validated against direct numerical simulation of the full pulse sequence to ~1e-8 (Table I). The headline constants h*=0.20683, n*=0.9295 m/(sigma_p k0 T), the crossover 0.91, and Delta_g_min = 4.3976 sigma_p/(mT sqrt N) are obtained by solving h'(a)=0 and elementary inequalities from the same model equations; no parameter is fitted to a target prediction, and no empirical data are used. There are no self-citations: Ref. [25] is an external result by another group, and its QFI is rederived inside the paper. A separate mathematical-scope concern exists: Eq. (19) is called a 'resolution-limited ceiling on fringe-based readout,' but it is optimized only over n at the mirrorless endpoint s=T_pi, whereas Section V optimizes asymmetry at fixed n. Within the same leading form, the joint limit u->0, n->infinity with beta u^2 fixed would make the fringe CFI diverge as 1/u^2, so the word 'ceiling' may overstate the model's content. That is a correctness/interpretation question, not circularity: the putative ceiling is not an input of the derivation, not a fitted parameter, and not a self-citation. Per the rules, non-circular correctness risks are excluded from the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; the model's physical inputs (source width, interrogation time, momentum transfer, detector resolution) are treated as independent variables. The numerical constants are derived from the model equations, not adjusted to data. The only approximations are the Gaussian-model assumptions listed as axioms.

assumptions (5)
  • domain assumption The beam-splitting pulses are instantaneous and lossless, realizing exactly the unitary in Eq. (1).
    Invoked in Sec. II and Eq. (1); real Bragg pulses have velocity-dependent efficiency and can populate higher orders or leave internal state unchanged.
  • domain assumption The motional initial state is a pure Gaussian with position variance σ²/2 and momentum variance δp² = ℏ²/(2σ²), with no interatomic interactions.
    Sec. II; the closed-form results rely on Gaussian integrals; interactions and mixed states would break the simple convolution structure.
  • domain assumption The detector response is a Gaussian convolution kernel of standard deviation σ_p.
    Sec. III and Eq. (7); real detectors include binning, background, and nonlinearities.
  • domain assumption The internal state remains two-level and the momentum transfer is exactly n photon momenta; the output ports are labeled by internal state.
    Sec. II; in Bragg diffraction the internal state is unchanged, so port labeling folds into the resolution budget.
  • standard math Standard quantum estimation theory: QFI/CFI definitions and Cramér–Rao bound.
    Sec. II, Eq. (4) and Refs. [23,24].

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Pith. "Pith review of Detection-resolution limits of large-momentum-transfer atom gravimetry." pith.science (2026). https://pith.science/paper/TNE6EGVW

@misc{pith2026260728749,
  author       = {Pith},
  title        = {Pith review of: Detection-resolution limits of large-momentum-transfer atom gravimetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TNE6EGVW}},
  note         = {Machine review of arXiv:2607.28749}
}
abstract

Large momentum transfer (LMT) enhances the gravitational phase of a light-pulse atom interferometer by a factor $n$, while mirrorless operation with momentum-resolved detection can quadruple the phase-carrying quantum Fisher information. These gains compete in practice because the momentum-space fringe period scales as $1/n$, making the fringe signal increasingly vulnerable to finite detector resolution. We analyze this trade-off in a solvable model with instantaneous lossless $n$-photon pulses, a Gaussian source, and Gaussian detection blur, allowing continuous interpolation between Kasevich--Chu and mirrorless geometries. Closed-form expressions for the blurred output distributions and classical Fisher information are obtained, with a fringe-phase averaging approximation whose error is exponentially suppressed and agrees with numerical simulations at the $10^{-8}$ level. We find that mirrorless operation surpasses a conventional interferometer with the same momentum transfer only when $\sigma_p < 0.91\,m/(n k_0 T)$. Fringe-based readout exhibits an optimal momentum transfer $n^* \simeq 0.93\,m/(\sigma_p k_0 T)$ and a resolution-limited sensitivity floor $\Delta g \simeq 4.4\,\sigma_p/(mT\sqrt{N})$, independent of photon momentum. When this criterion is not satisfied, partial mirror asymmetry can recover part of the enhancement, whereas population-based readout remains insensitive to detector blur and ultimately favors the conventional sequence at sufficiently large $n$. Estimates for $^{87}$Rb sensors show that the mirrorless advantage is primarily restricted to short-baseline instruments.

Figures

Figures reproduced from arXiv: 2607.28749 by the authors.

Figure 1
Figure 1. FIG. 1. Momentum distribution of output port [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. CFI of the mirrorless interferometer versus momen [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. CFI, normalized to the KC value [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

53 extracted references · 1 linked inside Pith

  1. [1]

    Figure 4 shows the family forn= 5: the closed form tracks the direct numerics through the interior optimum. BecauseR(e −x) = 1− √ 1−e −x = 1− √x+O(x 3/2) for smallx, the functionfis nonanalytic atu= 0, with the one-sided expansion (Appendix F F 3) f(u) = 1 + 4− p β u+ 2−4 p β u2 +O β3/2u3 ,(24) so an infinitesimal asymmetry increases the information whene...

  2. [2]

    Mo- mentum widths quoted below are in units ofℏk 0

    = 5.3µs. Mo- mentum widths quoted below are in units ofℏk 0. Ab- solute momentum resolution of order 10 −2 ℏk0 is rou- tine; reaching 10 −4 ℏk0 requires box-trapped or delta- kick-collimated condensates with long expansion, for which momentum spreads far below a single recoil have been observed [28], or spectroscopic velocimetry [27, 29]; momentum-resolve...

  3. [3]

    Momentum-space propagator In the momentum representation, ˆz=iℏ∂ p, and the Schr¨ odinger equationiℏ∂tψ= (ˆp2 z/2m+mgˆz)ψbecomes the first-order partial differential equation ∂tψ(p, t)−mg ∂pψ(p, t) =− ip2 2mℏ ψ(p, t).(A1) Along the characteristic curvesp(τ) =p+mg(t−τ), 0≤τ≤t(so thatp(t) =pandp(0) =p+mgt), Eq. (A1) reduces to the ordinary differential equa...

  4. [4]

    Action of the pulses The kick operators act in the momentum represen- tation as boosts, e±ink0 ˆzψ (p) =ψ(p∓nℏk 0), which follows frome ±ink0 ˆz|p⟩=|p±nℏk 0⟩. Writing the two- component motional spinor as (ψa, ψb), the pulses (1) act as ˆUπ/2 :ψ a(p)→ 1√ 2 ψa(p)−i ψb(p+nℏk 0) , ψb(p)→ 1√ 2 ψb(p)−i ψa(p−nℏk 0) ,(A4) ˆUπ :ψ a(p)→ −i ψb(p+nℏk 0), ψb(p)→ −i ψ...

  5. [5]

    The initial state isψ (0) a (p) =ψ 0(p),ψ (0) b = 0

    Composition of the full sequence For the composition we employ natural unitsℏ=m= k0 = 1 (momentum inℏk 0, time int 0 =m/ℏk 2 0, length ink −1 0 ); dimensions are restored in all final results. The initial state isψ (0) a (p) =ψ 0(p),ψ (0) b = 0. Applying Eq. (A4), then the propagator (A2) for timeT1, Eq. (A5), the propagator forT 2, and Eq. (A4) again, ea...

  6. [6]

    (A3) by elemen- tary algebra

    Branch phases The phase differences follow from Eq. (A3) by elemen- tary algebra. For porta, Φa,1 −Φ a,2 = Θ(p+gT 2, T1)−Θ(p+gT 2 +n, T1) + Θ(p+n, T2)−Θ(p, T2) =− T1 2 n 2(p+gT 2) +n − gnT 2 1 2 + T2 2 n(2p+n) + gnT 2 2 2 =−n(T 1 −T 2)p− ng 2 T 2 −2T 2 2 − n2(T1 −T 2) 2 ,(A10) where the last line usesT 2 1 + 2T1T2 −T 2 2 = (T1 +T 2)2 − 2T 2 2 =T 2 −2T 2 2...

  7. [7]

    Pure-state QFI from the fidelity Let|ψ g⟩be a normalized pure state depending smoothly ong. Expanding|ψ g+ϵ⟩=|ψ⟩+ϵ|∂ gψ⟩+ ϵ2 2 |∂2 g ψ⟩+O(ϵ 3), ⟨ψ|ψg+ϵ⟩= 1 +ϵ⟨ψ|∂ gψ⟩+ ϵ2 2 ⟨ψ|∂ 2 g ψ⟩+O(ϵ 3).(B1) Differentiating⟨ψ|ψ⟩= 1 once gives Re⟨ψ|∂ gψ⟩= 0, and twice gives Re⟨ψ|∂ 2 g ψ⟩=−⟨∂ gψ|∂gψ⟩. Hence ⟨ψ|ψg+ϵ⟩ 2 = 1−ϵ 2 h ⟨∂gψ|∂gψ⟩ − ⟨ψ|∂gψ⟩ 2i +O(ϵ 3) ≡1− FQ ϵ2...

  8. [8]

    Note that additive c-numbers in ˜ˆGdrop out of the variance

    Generator representation For|ψ g⟩= ˆU(g)|ψ in⟩with|ψ in⟩independent ofg, de- fine the Hermitian generator ˜ˆG≡i ˆU † ∂g ˆU .(B3) Then|∂ gψg⟩=−i ˆU ˜ˆG|ψin⟩, so⟨∂ gψ|∂gψ⟩=⟨ ˜ˆG2⟩in and ⟨ψ|∂gψ⟩=−i⟨ ˜ˆG⟩in, whence FQ = 4 Varin ˜ˆG ,(B4) the variance taken in theinputstate. Note that additive c-numbers in ˜ˆGdrop out of the variance

Show all 53 references
  1. [9]

    These follow fromdˆz H /ds= ˆp z,H /manddˆp z,H /ds= −mg: ˆpz,H (s) = ˆpz −mgsand ˆzH (s) = ˆz+ ˆpzs/m−gs 2/2

    Generator of a free-fall segment For ˆUg(t) =e −i ˆHt/ℏ with ˆH= ˆp 2 z/2m+mgˆz, the parameter-differentiation (Duhamel) identity ∂ge−i ˆHt/ℏ =− i ℏ Z t 0 e−i ˆH(t−s)/ℏ ∂g ˆH e−i ˆHs/ℏ ds(B5) with∂ g ˆH=mˆzgives ˜ˆGf (t) =i ˆU † g ∂g ˆUg = m ℏ Z t 0 ˆzH (s)ds,(B6) where ˆzH (s...

  2. [10]

    The chain rule applied to Eq

    Pulse conjugations and assembly Write the full sequence as ˆU= ˆK3 ˆUg(T2) ˆK2 ˆUg(T1) ˆK1 with ˆK1 = ˆK3 = ˆUπ/2 and ˆK2 = ˆUπ, allgindependent. The chain rule applied to Eq. (B3) gives ˜ˆG= ˆK † 1 ˜ˆGf (T1) ˆK1 + ˆK † 1 ˆU † g (T1) ˆK † 2 ˜ˆGf (T2) ˆK2 ˆUg(T1) ˆK1. (B8) 10 T...

  3. [11]

    (B12): ˆK † 1 ˆpz ˆK1 = 1 2 (1 +i ˆΣ1)ˆpz(1− i ˆΣ1) = 1 2 [ˆpz −i[ˆpz, ˆΣ1] + ˆΣ1 ˆpz ˆΣ1], and the algebra (B10) gives the stated result; for Eq

    Using [ˆpz, e±ink0 ˆz] =±nℏk 0 e±ink0 ˆz, direct computation gives ˆΣ1 ˆpz ˆΣ1 = ˆpz −nℏk 0 ˆσ,[ˆp z, ˆΣ1] =i nℏk0 ˆΣ2, ˆΣ1 ˆσˆΣ1 =−ˆσ, i[ ˆΣ1,ˆσ] = 2ˆΣ2,(B10) from which the three needed conjugation identities fol- low: ˆK † 2 ˆpz ˆK2 = ˆΣ1 ˆpz ˆΣ1 = ˆpz −nℏk 0 ˆσ,(B11) ˆK † ...

  4. [12]

    (A7)–(A9), by inserting it into Eq

    Independent check from the branch amplitudes The same result follows directly from the output state, Eqs. (A7)–(A9), by inserting it into Eq. (4). Writing (natural units)χ s(p) =c s ψ0(p+gT−q s) P j εsjeiΦsj (p,g) withq a = 0,q b =n, thegderivative produces an envelope-shift t...

  5. [13]

    Then, with Σ 2 =δp 2 +σ 2 p, Z ∞ −∞ dq Gσp (p−q)Q(q)e iκq =Q Σ(p)e iκp δp2/Σ2 ×e −κ2σ2 pδp2/2Σ2 ,(D1) whereQ Σ(p) =e −p2/2Σ2 / √ 2πΣ2

    Convolution lemma Lemma.LetQ(q) =e −q2/2δp2 / p 2πδp 2 andG σp (q) = e−q2/2σ2 p / q 2πσ 2p. Then, with Σ 2 =δp 2 +σ 2 p, Z ∞ −∞ dq Gσp (p−q)Q(q)e iκq =Q Σ(p)e iκp δp2/Σ2 ×e −κ2σ2 pδp2/2Σ2 ,(D1) whereQ Σ(p) =e −p2/2Σ2 / √ 2πΣ2. Proof.The exponent of the integrand is − (p−q) 2 2...

  6. [14]

    Blurred port distributions Apply the lemma to Eq. (9). Writing the fringe as the real part ofQ(p+mgT)e i(κp+ϕ) withϕ=ϕ g ±ϕ r, substituteq ′ =q+mgTin the convolution integral: Z dq Gσp (p−q)Q(q+mgT)e iκq =e −iκmgT Z dq′ Gσp (p+mgT)−q ′ Q(q′)e iκq′ =C e QΣ(p+mgT)e i[κep+ (κe−κ)...

  7. [15]

    Decomposition of the CFI Write the blurred distributions (11) as ˜P± = 1 2 ˜Q(1± Ce cos ˜α) with ˜Q(p) =Q Σ(p+mgT) and ˜α(p) =κ ep+ ˜ϕg ±ϕ r. Then ∂g ˜P± = 1 2 h mT ˜Q′ 1±C e cos ˜α ∓ ˜Q Ce ∂g ˜ϕg sin ˜α i , (D5) and therefore ∂g ˜P± 2 ˜P± = m2T 2 2 ˜Q′2 ˜Q 1±C e cos ˜α ∓mT ˜Q...

  8. [16]

    (D8) is of the formR ˜Q(p)W ˜α(p) dpwithWperiodic of periodπ

    F ringe-phase average The integral of Eq. (D8) is of the formR ˜Q(p)W ˜α(p) dpwithWperiodic of periodπ. When the fringe phase winds through many periods across the envelope (quantified in D 5), the integral approaches the uniform average⟨W⟩ α times R ˜Q= 1. The required averag...

  9. [17]

    Error bound for the fringe average The sole approximation above is the replacement ofR ˜Q W( ˜α)dpby⟨W⟩α. Expanding the periodic function in its Fourier series,W(α) = P l ˆWl e2ilα with ˆW0 = ⟨W⟩ α, the error is X l̸=0 ˆWl Z ˜Q(p)e 2il(κep+const) dp ≤ X l̸=0 ˆWl e−2l2κ2 eΣ2 , ...

  10. [18]

    Leading form and its accuracy Forσ p ≪δp, expand Eqs. (10) and (12): Σ 2 =δp 2 [1 + σ2 p/δp2];C 2 e =C 2 exp[+κ2σ4 p/Σ2] =C 2 1+O(a σ2 p/δp2) witha=κ 2σ2 p; and∂ g ˜ϕg =∂ gϕg 1−O(σ 2 p/δp2) for the family (for the mirrorless case explicitly∂ g ˜ϕg = nk0T 2 2 (δp2 −σ 2 p)/Σ2). ...

  11. [19]

    (15) is Ffringe = n2k2 0T 4 4 R e−a , a= σ2 pn2k2 0T 2 m2 ,(E1) and eliminatingn 2k2 0T 2 =a m2/σ2 p gives Eq

    Reduction to a single variable and the fixed point For the mirrorless configuration (T 2 = 0, ∆z= nℏk0T /m) in the design regimeσp ≪δp, the fringe term of Eq. (15) is Ffringe = n2k2 0T 4 4 R e−a , a= σ2 pn2k2 0T 2 m2 ,(E1) and eliminatingn 2k2 0T 2 =a m2/σ2 p gives Eq. (16) wi...

  12. [20]

    Ceiling, floor, and crossover At the optimum,F C(n∗) = (m 2T 2/4σ2 p)h ∗ + m2T 2/δp2 ≃0.05171m 2T 2/σ2 p in the regime (21) where the fringe term dominates, which is Eq. (19). The Cram´ er–Rao bound then gives ∆gmin = 1p N FC(n∗) = 2σp mT √ N h∗ = 4.3976σ p mT √ N ,(E4) Eq. (2...

  13. [21]

    (22) withp ln(16/7) = 0.909219

    Equal-ncomparison criterion The mirrorless fringe information exceeds the KC value n2k2 0T 4 π iff 4R(C 2)>1: 1− p 1−C 2 > 1 4 ⇔1−C 2 < 9 16 ⇔C 2 > 7 16 ,(E5) 13 i.e.e −a >7/16, ora <ln(16/7), which is Eq. (22) withp ln(16/7) = 0.909219. At the optimum itself, 4R(a ∗) = 4(1−w ...

  14. [22]

    (15) is n2k2 0 4 T 2 −2T 2 2 2 R=n 2k2 0T 4 π 2−(1−u) 2 2 R,(F2) and with ∆z= 2nℏk 0s/m= 2nℏk 0Tπu/mthe contrast argument isσ 2 p∆z2/ℏ2 =βu 2 withβas in Eq

    Reduction tof(u) WithT= 2T π,s=uT π, andT 2 =T π(1−u), T 2 −2T 2 2 = 4T 2 π −2T 2 π (1−u) 2 = 2T 2 π 2−(1−u) 2 ,(F1) so the fringe term of Eq. (15) is n2k2 0 4 T 2 −2T 2 2 2 R=n 2k2 0T 4 π 2−(1−u) 2 2 R,(F2) and with ∆z= 2nℏk 0s/m= 2nℏk 0Tπu/mthe contrast argument isσ 2 p∆z2/ℏ...

  15. [23]

    Stationarity condition Withx(u) =e −βu2 ,dR/dx= 1/(2 √1−x) and dx/du=−2βux, sodR/du=−βux/ √1−xandf ′(u) = 0 reads 4(1−u) 2−(1−u)2 R(x) = 2−(1−u)2 2 βu x√1−x ,(F3) a transcendental equation solved numerically for the op- tima shown in Fig. 4

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    The pref- actor is a polynomial, 2−(1−u) 2 2 = (1 + 2u−u 2)2 = 1 + 4u+ 2u 2 −4u 3 +u 4

    Nonanalytic expansion and the near-threshold optimum For smalluat fixedβ, expand both factors. The pref- actor is a polynomial, 2−(1−u) 2 2 = (1 + 2u−u 2)2 = 1 + 4u+ 2u 2 −4u 3 +u 4. For the contrast factor, 1−x=βu 2 − 1 2 β2u4 +O(β 3u6), so √ 1−x= p β|u| h 1− βu2 4 +O(β 2u4) ...

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