Pith. sign in

REVIEW 3 major objections 6 minor 1 cited by

Squeezing-enhanced accurate differential sensing under large phase noise

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Spin-squeezed states in both interferometers let differential phase sensing beat the standard quantum limit even when common-mode phase noise covers the full $2\pi$ range.

desk verdict The sensitivity scaling N^{-2/3} is real and worth building on, but the claimed N^{-4/3} bias suppression is likely an artifact of dropping the second-order delta-method term. read the letter →

arxiv 2501.18256 v1 pith:XJREP2MW submitted 2025-01-30 quant-ph

Robin Corgier , Marco Malitesta , Leonid A. Sidorenkov , Franck Pereira Dos Santos , Gabriele Rosi , Guglielmo M. Tino , Augusto Smerzi , Leonardo Salvi
show 1 more author
Luca Pezzè
This is my paper · ORCID
classification quant-ph
keywords spinsqueezingatominterferometrydifferentialphaseestimationellipsefittingstandardquantumlimitcommon-modenoisemetrologyone-axistwisting
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

Two atom interferometers that share a noisy laser usually lose their squeezed-state advantage when common-mode phase noise randomizes the phase over a full circle. The paper claims that feeding both interferometers with spin-squeezed states of a particular strength $\tau_*$ restores it: estimating the differential phase by fitting the correlated outputs to an ellipse gives a phase uncertainty that scales as $N^{-2/3}$ per shot, a gain of $N^{1/6}$ over the standard quantum limit, while the fitting bias falls as $N^{-4/3}$ instead of the $1/N$ found for coherent states. The protocol requires no calibration of the output statistics and uses squeezed states that are already available in experiments. If correct, it would make entanglement useful under exactly the large common-mode noise that differential atom sensors are built to reject.

What carries the argument

The central machinery is the one-axis-twisted spin-squeezed state $|\psi_{\mathrm{Squ}}\rangle = e^{-i\nu\hat{J}_x}e^{-i\tau\hat{J}_z^2}|\psi_{\mathrm{Coh}}\rangle$ sent into each of two Ramsey interferometers, together with the ellipse-fitting estimator acting on the joint output distribution $P(z_A,z_B|\delta\phi)=\int_0^{2\pi} d\phi_{\mathrm{cn}}\,P_0(z_A|\phi_A)P_0(z_B|\phi_B)/(2\pi)$. The argument turns on the variance-balance condition $\sigma^2_z|_{\phi=0}=\sigma^2_z|_{\phi=\pi/2}$, which selects $\tau_*$, and on a first-order Taylor (delta-method) expansion of the one-parameter fit, whose cubic equation in $h=\cos\delta\phi$ yields the bias formula used to show $H_2=0$ at $\tau_*$. Four fitting variants are compared--trace-constrained algebraic, ellipse-specific algebraic, geometric, and one-parameter--and the geometric fit comes closest to the Fisher-information bound at small differential phase.

What would settle it

Simulate the distribution of Eq. (6) for two squeezed states at $\tau=\tau_*$, with atom number $N$ between $10^2$ and $10^4$ and many sampled points per ellipse, and estimate $\delta\phi$ with the trace-constrained and one-parameter fits. The claim fails if the per-shot uncertainty does not approach the $N^{-2/3}$ scaling relative to the SQL, or if the bias decays as $1/N$ rather than $1/N^{4/3}$ as $N$ grows. A direct check is to confirm that the empirical bias crosses zero near $\tau_*$ and that the crossing point converges to Eq. (18) as $N$ increases.

Watch

Extended reading notes

Core claim

With a common phase $\phi_{\mathrm{cn}}$ uniformly distributed over $[0,2\pi]$, the mean outputs of the two interferometers trace an ellipse whose shape encodes the differential phase $\delta\phi$. The paper shows that at the squeezing strength $\tau_* \simeq (2/N^5)^{1/6}$, set by equating the projection-noise variances at $\phi=0$ and $\phi=\pi/2$, the noise around that ellipse becomes phase-independent. At this strength, ellipse fitting extracts $\delta\phi$ with a per-shot standard deviation $\sigma_{\delta\phi} \sim N^{-2/3}$ and a bias whose leading term is $\sim N^{-4/3}$; the compact bias formula $B(\delta\phi_{\mathrm{est}}) \approx -4\cot\delta\phi\,(H_0+H_2 h^2)/(1+2h^2)$, with $h=\cos\delta\phi$, has $H_2=0$ at $\tau_*$, removing the dominant coherent-state bias. The resulting sensitivity lies within about 1.5 dB of the Cramér-Rao bound for the same states, and the $N^{1/6}$ quantum gain holds over the whole range $0\lesssim\delta\phi\lesssim\pi/2$ for the ellipse method, while a hybrid classical-sensor fringe fit remains complementary near $\delta\phi=0$.

Load-bearing premise

The bias and variance formulas assume that the sample moments feeding the cubic fit are close to their mean values, so a first-order Taylor expansion around the large-$N$, zero-squeezing limit captures the estimator's behaviour; with finite data the zero-bias point shifts away from $\tau_*$ and a residual bias remains.

Editorial extensions

If this is right

  • A differential gravimeter or gradiometer subjected to large vibration or laser-phase noise can gain a factor $N^{1/6}$ over the standard quantum limit by tuning the squeezing strength to $\tau_*$, with no calibration of the readout distribution.
  • The bias reduction means long averaging is not dominated by a systematic fitting offset: the leading bias scales as $N^{-4/3}$ instead of $1/N$.
  • Because the sensitivity at $\tau_*$ is essentially independent of the differential phase, the protocol works across the wide range $0\lesssim\delta\phi\lesssim\pi/2$.
  • Near $\delta\phi=0$ the hybrid classical-sensor fringe-fitting method remains the better choice, so the two approaches are complementary rather than interchangeable.

Reading between the lines

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

  • Inference: The finite-sample shift of the zero-bias point away from $\tau_*$ suggests an adaptive version of the protocol could estimate the optimal squeezing strength from early data and then hold it, trading some measurement time for lower bias.
  • Inference: Since the geometric fit already approaches the Cramér-Rao bound at small $\delta\phi$, a maximum-likelihood estimator on the same squeezed states may close the remaining 1.5 dB gap, at the price of the calibration step the paper avoids.
  • Inference: The variance-balance condition that defines $\tau_*$ suggests a design principle for other probe states: make the projection noise isotropic over the Bloch sphere, which could extend the scaling gain to other entangled or variational states if they become experimentally available.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This paper theoretically analyzes differential phase estimation with two atom interferometers using spin-squeezed states in the presence of large common-mode phase noise spanning the full 2π range. The authors propose model-free ellipse fitting as the estimator and derive an optimal squeezing strength τ* that equalizes the phase-dependent variance contributions. They claim that, at τ*, the single-point differential phase variance scales as N^{-2/3} (a gain of N^{1/6} over the SQL) and that the estimation bias scales as B ~ 1/N^{4/3}, compared with 1/N for coherent states. The analytical scaling laws are derived from first principles in Sec. II and the Appendix, and are compared with Monte Carlo simulations for several fitting algorithms, the Cramér-Rao bound, and a hybrid approach with a classical sensor.

Significance. If the claims hold, this is a valuable contribution to quantum-enhanced interferometry: it shows that spin squeezing can simultaneously improve precision and accuracy in a realistic large-noise scenario using readily available states and a calibration-free estimator. The paper provides concrete, falsifiable scaling predictions and benchmarks against the CRB, and it explicitly compares different fitting methods. The variance scaling N^{-2/3} and the corresponding gain N^{1/6} are well supported by both analytic calculations and simulations, making the core metrological result credible and of direct interest to the atom interferometry and quantum metrology communities.

major comments (3)
  1. [Sec. III C, Appendix B.b, Eq. (30)] The claimed bias scaling B ~ 1/N^{4/3} at τ = τ* is derived from a first-order delta-method expansion of the estimator f about the unbiased point g∞. This expansion retains only the mean-shift terms (∂f/∂G_l)(⟨G_l⟩ − g∞_l), while the second-order contribution (1/2)Σ_{j,l}(∂²f/∂G_j∂G_l) Cov(G_j, G_l) is omitted. Because the G_l are sample means over N independent measurements, Cov(G_j, G_l) = O(1/N), so the second-order term is generically O(1/N). At τ = τ* the first-order term is suppressed to O(1/N^{4/3}) via H0 ∼ σ_z² ∼ N^{-4/3} and H2 = 0, but unless the second-order term vanishes identically or is also subleading at τ*, which the manuscript does not demonstrate, the asymptotic bias is O(1/N), not O(1/N^{4/3}). This is load-bearing because the abstract's claim of 'eliminating bias' and the improved asymptotic accuracy relative to coherent states are central to the paper's contribution. The authors should compute or bound the second-order delta-method term, or verify the scaling with Monte Carlo at larger N using an estimator that does not rely on the first-order formula.
  2. [Fig. 5, Table I] The numerical evidence for the 1/N^{4/3} bias scaling is not conclusive. The power-law fits are restricted to the range 300 < N < 1000, and the one-parameter analytical curves shown as solid lines in Fig. 5 are obtained from the same first-order formula Eq. (27)/(30), so they do not independently confirm the exponent. The error bars on the smallest bias values are also large. To distinguish a true asymptotic 1/N^{4/3} tail from a crossover between a 1/N^{4/3} transient and a 1/N asymptotic term, the authors should extend the range of N (e.g., to 10^5 or 10^6) or provide an independent numerical evaluation of the bias, for instance by directly solving the cubic equation at each Monte Carlo realization without the delta-method approximation.
  3. [Abstract and Sec. IV] The phrase 'eliminating bias inherent in ellipse fitting methods' overstates the result, since the analysis itself shows a residual bias B(δϕ_est) ≠ 0 at τ = τ*, with a scaling that is at best N^{-4/3} (and possibly O(1/N) if the second-order term dominates). The conclusion in Sec. IV similarly claims that spin squeezing 'can remarkably suppress the bias' without specifying the residual scaling. The authors should rephrase to 'strongly suppress' and explicitly report the residual bias and its asymptotic behavior, keeping the claim consistent with the analysis.
minor comments (6)
  1. [Notation throughout] The symbol N is used both for the number of atoms per interferometer and for the number of measurement points per ellipse (e.g., Eq. (7) vs. Fig. 5). This is confusing, especially in Sec. II C and Appendix C. Consider introducing separate notation such as N_at for the atom number and N_pts for the sample size.
  2. [Eq. (15)] The expression σ_SQL_δφ = √2 N^{-1/2} N^{-1/2} appears to contain an extra factor N^{-1/2}; clarify which N is the atom number and which is the number of measurements, or combine them into a single symbol with an explicit definition.
  3. [Appendix B.b, Eq. (B10)] The shorthand notations σ²_0 and σ²_π/2 are introduced in the main text but the equations in Appendix B would benefit from a restatement of these definitions for self-containedness, particularly because the calculation in Eq. (B10) heavily relies on them.
  4. [Fig. 4] The legend in Fig. 4 refers to 'one parameter' while the caption calls it 'algebraic one-parameter'; unify the terminology to avoid confusion with the two-parameter algebraic fits.
  5. [Reference [33]] The reference has a typo: 'Pezz`e ans Smerzi' should be 'Pezzè and Smerzi'.
  6. [Appendix C, Fig. 7] The notation N is again used for the number of points in an ellipse while in the main text N denotes the atom number; please use distinct symbols and clarify the axes in Fig. 7.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the optimal squeezing strength is derived from a variance-equality condition, not from the target scaling.

full rationale

The paper's central results are self-contained derivations from the OAT spin-squeezed-state model. The optimal squeezing strength τ* is defined in Eq. (18) by the variance-equality condition σ^2_z|ϕ=0 = σ^2_z|ϕ=π/2, which is an objective minimization of the phase dependence of quantum projection noise, not a parameter fitted to reproduce the claimed N^{-2/3} or N^{-4/3} scalings. The sensitivity scaling σδϕ ~ N^{-2/3} follows by error propagation from the resulting phase-independent variance and the fringe contrast, and it is benchmarked against the SQL and the Cramér-Rao bound. The bias suppression is derived in Appendix B.b via a first-order delta-method expansion, yielding Eq. (30); the statement that H2 = 0 at τ = τ* is a consequence of the same variance-equality condition used to define τ*, so no fitted input is renamed as a prediction. Self-citations such as Ref. [25] are contextual and not load-bearing for the central differential-interferometer result. The possible omission of second-order delta-method terms in the bias expansion is a correctness/asymptotic-accuracy concern, not a circularity: the derivation does not assume the target bias exponent as an input.

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

The protocol depends on standard SU(2) interferometry, the uniform full-cycle common-noise assumption, ideal one-axis-twisting squeezed states with perfect readout, and the delta-method approximation for the fit statistics. No extra free parameters are fitted; the optimal squeezing tau* is derived from a variance-equality condition.

assumptions (6)
  • domain assumption The two interferometers are conditionally independent given the common phase phi_cn, so the joint output distribution factorizes as P(zA,zB|delta-phi) = integral P0(zA|phiA) P0(zB|phiB) P(phi_cn) d phi_cn.
    Central to Eq. (6); assumes the only shared randomness between the two interferometers is the common phase noise.
  • domain assumption The common phase noise is uniformly distributed over the full [0, 2*pi] interval.
    Defines the large-noise regime studied; the ellipse-tracing relies on full coverage of the Lissajous figure. Realistic noise may be narrower or colored.
  • domain assumption Probe states are ideal one-axis-twisting squeezed states with no loss or decoherence, and readout is a perfect projective measurement of zM.
    Used throughout; any inefficiency will degrade the gain and modify the optimal tau. The variance formulas Eqs. (13)-(14) assume this idealization.
  • standard math The SU(2) coherent-state moment generating function (Eq. B8) is valid for computing moments of the phase-shifted squeezed state.
    Used in Appendix B.a; standard spin-algebra result from Refs. [77,92].
  • standard math The estimator f is a smooth function of the sample moments, so the delta-method / first-order Taylor expansion applies for large N.
    Used for Eqs. (27)-(28) and the bias formula Eq. (30); relies on asymptotic normality of sample moments and requires a large number of measurement points per ellipse.
  • domain assumption For large atom number (N >= 10^4), the discrete output distribution can be approximated by a Gaussian with the same mean and variance.
    Used only for the scaling results in Fig. 6b; not central to the main claims.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Squeezing-enhanced accurate differential sensing under large phase noise." pith.science (2026). https://pith.science/paper/XJREP2MW

@misc{pith2026250118256,
  author       = {Pith},
  title        = {Pith review of: Squeezing-enhanced accurate differential sensing under large phase noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XJREP2MW}},
  note         = {Machine review of arXiv:2501.18256}
}
abstract

Atom interferometers are reaching sensitivities fundamentally constrained by quantum fluctuations. A main challenge is to integrate entanglement into quantum sensing protocols to enhance precision while ensuring robustness against noise and systematics. Here, we theoretically investigate differential phase measurements with two atom interferometers using spin-squeezed states, accounting for common-mode phase noise spanning the full $2\pi$ range. We estimate the differential signal using model-free ellipse fitting, a robust method requiring no device calibration and resilient to additional noise sources. Our results show that spin-squeezing enables sensitivities below the standard quantum limit. Specifically, we identify optimal squeezed states that minimize the differential phase variance, scaling as $N^{-2/3}$, while eliminating bias inherent in ellipse fitting methods. We benchmark our protocol against the Cram\'er-Rao bound and compare it with hybrid methods that incorporate auxiliary classical sensors. Our findings provide a pathway to robust and high-precision atom interferometry, in realistic noisy environments and using readily available states and estimation methods.

Figures

Figures reproduced from arXiv: 2501.18256 by the authors.

Figure 1
Figure 1. It consists of two Ramsey interferometers operating in parallel and using a common laser to generate beam splitters and mirrors (not explicitly shown in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Illustration of di [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. (c,d) shows the effective single-point differential phase sensitivity, σ eff δϕest , following the same color code and symbols as in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Bias, [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Ellipse fitting vs fringe fitting. In all panels, the config [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Bias [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Atomic Coherence of 2 minutes and Instability of 1.5E-18 at 1 s in a Wannier-Stark Lattice Clock

    physics.atom-ph 2025-05 conditional novelty 7.0 of 10

    A shallow-lattice 87Sr clock reaches roughly 118 s coherence time and 1.5e-18 fractional instability at 1 s, with decoherence traced to Raman scattering and spectator-atom collisions.

Reference graph

Works this paper leans on

96 extracted references · 77 canonical work pages · cited by 1 Pith paper

  1. [1]

    trace method

    Ellipse fitting and statistical analysis A fit of the measurement data aims at recovering the “true” ellipse in Eq. (17) and provides an estimate δϕest of the dif- ferential phase. The figures of merit we consider are both the variance σ2 δϕest =⟨δϕ2 est⟩−⟨ δϕest⟩2, (20) which quantifies the precision of the estimate, and the devia- tion of the mean estim...

  2. [2]

    butterfly-like shape

    We also introduce collective pseudo-spin oper- ators ˆJM,x = ( ˆm† 1 ˆm2 + ˆm† 2 ˆm1)/2, ˆJM,y = ( ˆm† 1 ˆm2− ˆm† 2 ˆm1)/2i and ˆJM,z = ( ˆm† 1 ˆm1− ˆm† 2 ˆm2)/2, where {m, M} = {a, A} or {b, B}. These operators satisfy the usual commutation rela- tions [ ˆJM,i, ˆJM, j] = iϵi jk ˆJM,k with ϵi jk being the Levi-Civita symbol. For simplicity, we assume that...

  3. [3]

    A. D. Cronin, J. Schmiedmayer and D. E. Pritchard, Optics and interferometry with atoms and molecules, Rev. Mod. Phys. 81, 1051 (2009)

    2009

  4. [4]

    G. M. Tino and M. A. Kasevich (editors) Atom Interferometry (SIF and IOS Press, Bologna, Amsterdam, 2014)

    2014

  5. [5]

    M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, Search for new physics with atoms and molecules, Rev. Mod. Phys. 90, 025008 (2018)

    2018

  6. [6]

    Geiger, A

    R. Geiger, A. Landragin, S. Merlet, and F. Pereira Dos Santos, High-accuracy inertial measurements with cold-atom sensors, AVS Quantum Sci. 2, 024702 (2020)

    2020

  7. [7]

    Bongs, M

    K. Bongs, M. Holynski, J. V ovrosh, P. Bouyer, G. Condon, E. Rasel, C. Schubert, W. P. Schleich, and A. Roura, Taking atom interferometric quantum sensors from the laboratory to real-world applications, Nat. Rev. Phys. 1, 731 (2019)

    2019

  8. [8]

    Gauguet, B

    A. Gauguet, B. Canuel, T. L ´ev`eque, W. Chaibi, and A. Landra- gin, Characterization and limits of a cold-atom Sagnac interfer- ometer, Phys. Rev. A 80 063604 (2009)

    2009

Show all 96 references
  1. [9]

    Sorrentino, Q

    F. Sorrentino, Q. Bodart, L. Cacciapuoti, Y .-H. Lien, M. Prevedelli, G. Rosi, L. Salvi, and G. M. Tino, Sensitivity limits of a Raman atom interferometer as a gravity gradiometer,Phys. Rev. A 89, 023607 (2014)

    2014

  2. [10]

    Janvier et al., Compact differential gravimeter at the quantum projection-noise limit, Phys

    C. Janvier et al., Compact differential gravimeter at the quantum projection-noise limit, Phys. Rev. A 105, 022801 (2022)

    2022

  3. [11]

    Pezz `e and A

    L. Pezz `e and A. Smerzi, Entanglement, nonlinear dynamics, and the Heisenberg limit, Phys. Rev. Lett. 102, 100401 (2009)

    2009

  4. [12]

    Pezz `e, A

    L. Pezz `e, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys. 90, 035005 (2018)

    2018

  5. [13]

    Kitagawa and M

    M. Kitagawa and M. Ueda, Squeezed spin states, Phys. Rev. A 47, 5138 (1993)

    1993

  6. [14]

    D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, Squeezed atomic states and projection noise in spectroscopy, Phys. Rev. A 50, 67 (1994)

    1994

  7. [15]

    J. Ma, X. Wang, C.P. Sun and F. Nori, Quantum spin squeezing, Phys. Rep. 509, 89 (2011)

    2011

  8. [16]

    Louchet-Chauvet, J

    A. Louchet-Chauvet, J. Appel, J. J. Renema, D. Oblak, N. Kjaergaard, and E. S. Polzik, Entanglement-assisted atomic clock beyond the projection noise limit,New J. Phys.12 065032 (2010)

    2010

  9. [17]

    Kruse, K

    I. Kruse, K. Lange, J. Peise, B. L ¨ucke, L. Pezz `e, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt, Im- 10 provement of an atomic clock using squeezed vacuum, Phys. Rev. Lett. 117, 143004 (2016)

    2016

  10. [18]

    Hosten, et al., Measurement noise 100 times lower than the quantum-projection limit using entangled atoms, Nature 529, 505 (2016)

    O. Hosten, et al., Measurement noise 100 times lower than the quantum-projection limit using entangled atoms, Nature 529, 505 (2016)

    2016

  11. [19]

    Laudat, et al., Spontaneous spin squeezing in a rubidium BEC, New J

    T. Laudat, et al., Spontaneous spin squeezing in a rubidium BEC, New J. Phys. 20, 073018 (2018)

    2018

  12. [20]

    Pedrozo-Penafiel, et al., Entanglement on an optical atomic- clock transition, Nature 588, 414 (2020)

    E. Pedrozo-Penafiel, et al., Entanglement on an optical atomic- clock transition, Nature 588, 414 (2020)

    2020

  13. [21]

    Huang et al., Observing Spin-Squeezed States under Spin- Exchange Collisions for a Second, PRX Quantum 4, 020322 (2023)

    M. Huang et al., Observing Spin-Squeezed States under Spin- Exchange Collisions for a Second, PRX Quantum 4, 020322 (2023)

    2023

  14. [22]

    J. M. Robinson, M. Miklos, Y . M. Tso, C. J. Kennedy, T. Both- well, D. Kedar, J. K. Thompson and J. Ye, Direct comparison of two spin-squeezed optical clock ensembles at the 10−17 level, Nat. Phys. 20, 208 (2024)

    2024

  15. [23]

    Salvi, N

    L. Salvi, N. Poli, V . Vuleti´c, and G M. Tino, Squeezing on Mo- mentum States for Atom Interferometry Phys. Rev. Lett. 120, 033601 (2018)

    2018

  16. [24]

    S. S. Szigeti et al., High-Precision Quantum-Enhanced Gravimetry with a Bose-Einstein Conden-sate, Phys. Rev. Lett. 125, 100402 (2020)

    2020

  17. [25]

    Corgier, L

    R. Corgier, L. Pezz `e and A. Smerzi, Nonlinear Bragg interfer- ometer with a trapped Bose-Einstein condensate, Phys. Rev. A 103, L061301 (2021)

    2021

  18. [26]

    Corgier, N

    R. Corgier, N. Gaaloul, A. Smerzi and L. Pezz `e, Delta-kick Squeezing, Phys. Rev. Lett. 127, 183401 (2021)

    2021

  19. [27]

    Corgier, M

    R. Corgier, M. Malitesta, A. Smerzi and L. Pezz `e, Quantum- enhanced di fferential atom interferometers and clocks with spin-squeezing swapping, Quantum 7, 965 (2023)

    2023

  20. [28]

    G. P. Greve, C. Luo, B. Wu, and J. K. Thompson, Entanglement-Enhanced Matter-Wave Interferometry in a High-Finesse Cavity, Nature 610, 472 (2022)

    2022

  21. [29]

    Malia et al., Distributed quantum sensing with mode- entanglement spin-squeezed atomic states, Nature 612, 661 (2022)

    B. Malia et al., Distributed quantum sensing with mode- entanglement spin-squeezed atomic states, Nature 612, 661 (2022)

    2022

  22. [30]

    Anders et al., Momentum Entanglement for Atom Interfer- ometry, Phys

    F. Anders et al., Momentum Entanglement for Atom Interfer- ometry, Phys. Rev. Lett. 127, 140402 (2021)

    2021

  23. [31]

    Cassens, B

    C. Cassens, B. Meyer-Hoppe, E.M Rasel, and C. Klempt, Entanglement-enhanced atomic gravimeter, arXiv:2404.18668

    arXiv

  24. [32]

    Andr `e, A

    A. Andr `e, A. S. Sørensen, and M. D. Lukin, Stability of Atomic Clocks Based on Entangled Atoms, Phys. Rev. Lett.92, 230801 (2004)

    2004

  25. [33]

    Braverman, A

    B. Braverman, A. Kawasaki, and V . Vuleti ´c, Impact of non- unitary spin squeezing on atomic clock performance, New. J. Phys. 20, 103019 (2018)

    2018

  26. [34]

    Pezz `e ans A

    L. Pezz `e ans A. Smerzi, Heisenberg-Limited Noisy Atomic Clock Using a Hybrid Coherent and Squeezed State Protocol, Phys. Rev. Lett. 125, 210503 (2020)

    2020

  27. [35]

    Pezz `e ans A

    L. Pezz `e ans A. Smerzi, Quantum Phase Estimation Algorithm with Gaussian Spin States, Phys. Rev. X Quantum 2, 040301 (2013)

    2013

  28. [36]

    Borregaard and A

    J. Borregaard and A. S. Sørensen, Near-Heisenberg- Limited Atomic Clocks in the Presence of Decoherence,Phys. Rev. Lett. 111, 090801 (2013)

    2013

  29. [37]

    Kaubruegger, D

    R. Kaubruegger, D. V . Vasilyev, M. Schulte, K. Hammerer, and P. Zoller, Quantum Variational Optimization of Ramsey Inter- ferometry and Atomic Clocks, Phys. Rev. X 11, 041045 (2021)

    2021

  30. [38]

    C. D. Marciniak, T. Feldker, I. Pogorelov, R. Kaubruegger, D. V . Vasilyev, R. van Bijnen, P. Schindler, P. Zoller, R. Blatt, and T. Monz, Optimal metrology with programmable quantum sen- sors, Nature 603, 604 (2022)

    2022

  31. [39]

    M. J. Snadden, J. M. McGuirk, P. Bouyer, K. G. Haritos, M. A. Kasevich, Measurement of the Earth’s gravity gradient with an atom interferometer-based gravity gradiometer,Phys. Rev. Lett. 81 971 (1998)

    1998

  32. [40]

    G. T. Foster, J. B. Fixler, J. M. McGuirk, and M. A. Kasevich, Method of phase extraction between coupled atom interferom- eters using ellipse-specific fitting, Opt. Lett. 27, 951 (2002)

    2002

  33. [41]

    Pereira dos Santos, Di fferential phase extraction in an atom gradiometer, Phys

    F. Pereira dos Santos, Di fferential phase extraction in an atom gradiometer, Phys. Rev. A. 91, 063615 (2015)

    2015

  34. [42]

    G. Rosi, L. Cacciapuoti, F. Sorrentino, M. Menchetti, M. Prevedelli, and G. M. Tino, Measurement of the Gravity-Field Curvature by Atom Interferometry,Phys. Rev. Lett.114, 013001 (2015)

    2015

  35. [43]

    Barrett, L

    B. Barrett, L. Antoni-Micollier, L. Chichet, B. Battelier, T. L´ev`eque, A. Landragin, and P. Bouyer, Dual Matter-Wave In- ertial Sensors in Weightlessness, Nature Communications , 7 2041-1723 (2016)

    2016

  36. [44]

    Langlois, R

    M. Langlois, R. Caldani, A. Trimeche, S. Merlet, F. Pereira Dos Santos, Differential phase extraction in dual interferometers ex- ploiting the correlation between classical and quantum sensors, Phys. Rev. A. 96, 053624 (2017)

    2017

  37. [45]

    R. H. Parker, C. Yu, W. Zhong, B. Estey, and H. M ¨uller, Mea- surement of the fine-structure constant as a test of the Standard Model, Science 360, 191 (2018)

    2018

  38. [46]

    G. Rosi, G. D’Amico, L. Cacciapuoti, F. Sorrentino, M. Prevedelli, M. Zych, ˇC. Brukner, and G. M. Tino, Quantum test of the equivalence principle for atoms in coherent superposition of internal energy states, Nat. Commun. 8, 15529 (2019)

    2019

  39. [47]

    L. Hu, N. Poli, L. Salvi, and G. M. Tino, Atom Interferome- try with the Sr Optical Clock Transition, Phys. Rev. Lett. 119, 263601 (2017)

    2017

  40. [48]

    Salvi, L

    L. Salvi, L. Cacciapuoti, G. M. Tino, and G. Rosi, Atom In- terferometry with Rb Blue Transitions, Phys. Rev. Lett. 131, 103401 (2023)

    2023

  41. [49]

    E. R. Elliott, et al. Quantum gas mixtures and dual-species atom interferometry in space, Nature 623, 502–508 (2023)

    2023

  42. [50]

    J. B. Fixler, G. T. Foster, J. M. McGuirk, and M. A. Kasevich, Atom Interferometer Measurement of the Newtonian Constant of Gravity, Science 5 74 (2007)

    2007

  43. [51]

    Lamporesi1, A

    G. Lamporesi1, A. Bertoldi, L. Cacciapuoti, M. Prevedelli, and G. M. Tino, Determination of the Newtonian Gravitational Con- stant Using Atom Interferometry, Phys. Rev. Lett. 100 050801 (2008)

    2008

  44. [52]

    G. Rosi, F. Sorrentino, L. Cacciapuoti, M. Prevedelli, and G. M. Tino, Precision measurement of the Newtonian gravitational constant using cold atoms, Nature 510, 518 (2014)

    2014

  45. [53]

    Overstreet, P

    C. Overstreet, P. Asenbaum, J. Curti, M. Kim, M. A. Kasevich, Observation of a gravitational Aharonov-Bohm e ffect Science 375, 226 (2022)

    2022

  46. [54]

    Dimopoulos, P

    S. Dimopoulos, P. W. Graham, J. M. Hogan, M. A. Kasevich, Testing general relativity with atom interferometry, Phys. Rev. Lett. 98 111102 (2007)

    2007

  47. [55]

    Varoquaux, R

    G. Varoquaux, R. A. Nyman, R. Geiger, P. Cheinet, A. Landra- gin, and P. Bouyer, How to estimate the di fferential accelera- tion in a two-species atom interferometer to test the equivalence principle, New J. Phys. 11 113010 (2009)

    2009

  48. [56]

    Canuel et al., ELGAR–a European Laboratory for Gravita- tion and Atom-interferometric Research,Class

    B. Canuel et al., ELGAR–a European Laboratory for Gravita- tion and Atom-interferometric Research,Class. Quantum Grav. 37 225017 (2020)

    2020

  49. [57]

    Dimopoulos, P

    S. Dimopoulos, P. W. Graham, J. M. Hogan, M. A. Kasevich, S. Rajendran, Atomic gravitational wave interferometric sensor, Phys. Rev. D 78 122002 (2008)

    2008

  50. [58]

    Asenbaum, C

    P. Asenbaum, C. Overstreet, M. Kim, J. Curti, M. A. Kasevich, Atom-Interferometric Test of the Equivalence Principle at the 10−12 Level, Phys. Rev. Lett. 125, 191101 (2020). 11

    2020

  51. [59]

    Struckmann, R

    C. Struckmann, R. Corgier, S. Loriani, G. Kleinsteinberg, N. Gox, E. Giese, G. M´etris, N. Gaaloul, P. Wolf, Platform and en- vironment requirements of a satellite quantum test of the Weak Equivalence Principle at the 10 −17 level, Phys. Rev. D 109, 064010 (2024)

    2024

  52. [60]

    Schlippert, J

    D. Schlippert, J. Hartwig, H. Albers, L.L. Richardson, C. Schu- bert, A. Roura, W.P. Schleich, W. Ertmer, and E.M. Rasel, Quantum test of the universality of free fall, Phys. Rev. Lett. 112, 203002 (2014)

    2014

  53. [61]

    Lin Zhou, Shitong Long, Biao Tang, Xi Chen, Fen Gao, Wencui Peng, Weitao Duan, Jiaqi Zhong, Zongyuan Xiong, Jin Wang, Yuanzhong Zhang, and Mingsheng Zhan, Test of equivalence principle at 10−8 level by a dual-species double-diffraction Ra- man atom interferometer, Phys. Rev. L...

    2015

  54. [62]

    Ke Zhang, Min-Kang Zhou, Yuan Cheng, Le-Le Chen, Qin Luo, Wen-Jie Xu, Lu-Shuai Cao, Xiao-Chun Duan, and Zhong- Kun Hu, Testing the universality of free fall by comparing the atoms in different hyperfine states with Bragg diffraction, Chin. Phys. Lett. 37, 043701 (2020)

    2020

  55. [63]

    Barrett, G

    B. Barrett, G. Condon, L. Chichet, L. Antoni-Micollier, R. Arguel, M. Rabault, C. Pelluet, V . Jarlaud, A. Landragin, P. Bouyer, and B. Battelier, Testing the universality of free fall using correlated atom interferometers, AVS Quantum Sci. 4, 014401 (2022)

    2022

  56. [64]

    Durfee D S, Shaham Y K and Kasevich M A Phys. Rev. Lett. 97, 240801 (2006)

    2006

  57. [65]

    Stray, et al., Quantum sensing for gravity cartography, Na- ture 602, 590 (2022)

    B. Stray, et al., Quantum sensing for gravity cartography, Na- ture 602, 590 (2022)

    2022

  58. [66]

    A. W. Young, W. J. Eckner, W. R. Milner, D. Kedar, M. A. Norcia, E. Oelker, N. Schine, J. Ye, and A, M. Kaufman, Half- minute-scale atomic coherence and high relative stability in a tweezer clock, Nature 588, 408 (2020)

    2020

  59. [67]

    Zheng, J

    X. Zheng, J. Dolde, V . Lochab, B. N. Merriman, H. Li, and S. Kolkowitz, Differential clock comparisons with a multiplexed optical lattice clock, Nature 602, 425 (2022)

    2022

  60. [68]

    M. J. Collett, and G. J. Tee, Ellipse fitting for interferometry. Part 2: experimental realization, Appl. Opt. 53, 7697 (2014)

    2014

  61. [69]

    M. J. Collett, and G. J. Tee, Ellipse fitting for interferometry. Part 3: dynamic method, J. Opt. Soc. Am. A 32, 491 (2015)

    2015

  62. [70]

    Stockton, Xinan Wu, and Mark A

    John K. Stockton, Xinan Wu, and Mark A. Kasevich, Bayesian estimation of differential interferometer phase, Phys. Rev. A76, 033613 (2007)

    2007

  63. [71]

    Eckert, P

    K. Eckert, P. Hyllus, D. Bruß, U. V . Poulsen, M. Lewenstein, C. Jentsch, T. M ¨uller, E. M. Rasel and W. Ertmer, Di fferential atom interferometry beyond the standard quantum limit, Phys. Rev. A 73, 013814 (2006)

    2006

  64. [72]

    Gessner, L

    M. Gessner, L. Pezz `e, and A. Smerzi, Sensitivity Bounds for Multiparameter Quantum Metrology, Phys. Rev. Lett. 121, 130503 (2018)

    2018

  65. [73]

    Landini, M

    M. Landini, M. Fattori, L. Pezz `e, and A. Smerzi, Phase- noise protection in quantum-enhanced di fferential interferom- etry New J. of Phys. 16, 113074 (2018)

    2018

  66. [74]

    M. J. Collett, and G. J. Tee, Ellipse fitting for interferometry. Part 1: static methods, J. Opt. Soc. Am. A 31, 2573 (2014)

    2014

  67. [75]

    H.-Y . Zhu, M. Xue, J. Li and M. Deng, E fficacy of di fferen- tial phase extraction methods in dual atom interferometers un- der correlated noise conditions, Results in Physics 52, 106855 (2023)

    2023

  68. [76]

    Zhang, J

    X. Zhang, J. Zhong, W. Lyu, W. Xu, L. Zhu, M. Wang, X. Chen, B. Tang, J. Wang, and M. Zhan, Dependence of the ellipse fit- ting noise on the di fferential phase between interferometers in atom gravity gradiometers, Opt. Express 31, 44102 (2023)

    2023

  69. [77]

    Ridley and A

    K. Ridley and A. Rodgers, An investigation of errors in ellipse- fitting for cold-atom interferometers, EPJ Quantum Technol. 11, 79 (2024)

    2024

  70. [78]

    Yurke, S

    B. Yurke, S. L. McCall, J. R. Klauder, SU (2) and SU (1, 1) interferometers Phys. Rev. A 33, 4033 (1986)

    1986

  71. [79]

    F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Atomic Coherent States in Quantum Optics, Phys. Rev. A 6, 2211 (1972)

    1972

  72. [80]

    We use σ2 z|ϕ=0≈ 1/(2N2τ2) + N2τ4/12 and σ2 z|ϕ=π/2≈ N3τ4/4, giving Eq

    An exact analytical solution to σ2 z|ϕ=0 = σ2 z|ϕ=π/2 can be ob- tained in the regime 1/N≪ τ≪ 1/ √ N, with N≫ 1. We use σ2 z|ϕ=0≈ 1/(2N2τ2) + N2τ4/12 and σ2 z|ϕ=π/2≈ N3τ4/4, giving Eq. (18)

  73. [81]

    [88] is the al- gebraic distance with different weights used on different points of the data set

    Other distances can be usually regarded as variations on these two: for example, the statistical distance of Ref. [88] is the al- gebraic distance with different weights used on different points of the data set

  74. [82]

    Gander, G.H

    W. Gander, G.H. Golub, and R. Strebel, Least-squares fitting of circles and ellipses, BIT 34, 558 (1994)

    1994

  75. [83]

    Rosin, A Note on the Least Squares Fitting of Ellipses, Pattern Recognition Letters 14, 799 (1993)

    P.L. Rosin, A Note on the Least Squares Fitting of Ellipses, Pattern Recognition Letters 14, 799 (1993)

    1993

  76. [84]

    Fitzgibbon, M

    A. Fitzgibbon, M. Pilu and R. B. Fisher, Direct least square fitting of ellipses,Proceedings of 13th International Conference on Pattern Recognition 1, 253 (1996)

    1996

  77. [85]

    Hal ´ır, J

    R. Hal ´ır, J. Flusser, Numerically stable direct least squares fit- ting of ellipses. Proceedings of the 6th International Confer- ence in Central Europe on Computer Graphics and Visualiza- tion. WSCG ’98. (1998)

    1998

  78. [86]

    Gander, Least squares with a quadratic constraint, Numer

    W. Gander, Least squares with a quadratic constraint, Numer. Math. 36, 291 (1980)

    1980

  79. [87]

    F. L. Bookstein, Fitting conic sections to scattered data, Com- puter Graphics and Image Processing 9(1), 56 (1979)

    1979

  80. [88]

    Taubin, Estimation of Planar Curves, Surfaces and NonPla- nar Space Curves Defined by Implicit Equations, With Applica- tions to Edge and Range Image Segmentation,IEEE Trans

    G. Taubin, Estimation of Planar Curves, Surfaces and NonPla- nar Space Curves Defined by Implicit Equations, With Applica- tions to Edge and Range Image Segmentation,IEEE Trans. Pat- tern Analysis and Machine Intelligence 13 (11), 1115 (1991)

    1991

  81. [89]

    Fitzgibbon, and R.B

    A.W. Fitzgibbon, and R.B. Fisher, A Buyer’s Guide to Conic Fitting, Proceedings of the British Machine Conference. BMVC 1995 (1995)

    1995

  82. [90]

    Kanatani, Statistical bias of conic fitting and renormaliza- tion, IEEE Transactions on Pattern Analysis and Machine In- telligence 16(3), 320 (1994)

    K. Kanatani, Statistical bias of conic fitting and renormaliza- tion, IEEE Transactions on Pattern Analysis and Machine In- telligence 16(3), 320 (1994)

    1994

  83. [91]

    Buˇzek, R

    V . Buˇzek, R. Derka, and S. Massar, Optimal Quantum Clocks, Phys. Rev. Lett. 82, 2207 (1999)

    1999

  84. [92]

    H. P. Gavin, The Levenberg-Marquardt algorithm for nonlinear least squares curve-fitting problems, Department of Civil and Environmental Engineering Duke University August 3(2019)

    2019

  85. [93]

    Hurt, Asymptotic expansions of functions of statistics, Ap- likace matematiky 21, 444 (1976)

    J. Hurt, Asymptotic expansions of functions of statistics, Ap- likace matematiky 21, 444 (1976)

    1976

  86. [94]

    R. R. Puri, Mathematical Methods of Quantum Optics (Springer-Verlag Berlin, Heidelberg, 2001). 12 V . APPENDIX Appendix A: Conic fitting In this section we provide an overview of the ellipse-fitting methods that are relevant to the present work, see Table II. We will restrict ...

    2001

  87. [95]

    (22) is defined as dG(zA, j, zB, j; C) = min (zA,zB)∈C q (zA− zA, j)2 + (zB− zB, j)2

    Geometric fit The geometric distance of the point ( zA, j, zB, j) from the conic C of Eq. (22) is defined as dG(zA, j, zB, j; C) = min (zA,zB)∈C q (zA− zA, j)2 + (zB− zB, j)2. (A1) This optimization problem requires to compute the roots of a fourth-order polynomial [72] which ...

  88. [96]

    (22) is defined as dAL(zA, j, zB, j; C) = kT j v, (A2) where k j = (z2 A, j, zA, jzB, j, z2 B, j, zA, j, zB, j, 1)T

    Algebraic fit The algebraic distance of the point (zA, j, zB, j) from the conic C of Eq. (22) is defined as dAL(zA, j, zB, j; C) = kT j v, (A2) where k j = (z2 A, j, zA, jzB, j, z2 B, j, zA, j, zB, j, 1)T . Equation (A2) is linear with respect to v, making the algebraic fit a ...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.