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REVIEW 3 major objections 4 minor 107 references

Heisenberg scaling in optical magnetometry with measurement-induced correlations as a quantum resource

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Pith's one-line read This paper argues that the continuous collective measurement of an atomic ensemble in an optical magnetometer can itself generate the correlations needed for Heisenberg scaling of the quantum Fisher information, in a dissipative steady stat

desk verdict The semiclassical CRB-violation diagnostic and the full-counting machinery are real; the Heisenberg-scaling headline collapses against the paper's own thermodynamic-limit scaling. read the letter →

arxiv 2601.01820 v2 pith:FBZFCCBU submitted 2026-01-05 quant-ph

Georg Engelhardt , Ming Li , Xingchang Wang , JunYan Luo , J.F. Chen This is my paper · ORCID
classification quant-ph
keywords opticalmagnetometryquantumFisherinformationHeisenbergscalingmeasurement-inducedcorrelationsCramér-Raoboundcontinuousmeasurementcollectivespinfullcountingstatistics
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 aims to show that in an optical magnetometer, the continuous collective measurement of light passing through a hot atomic vapor can itself create the many-body quantum correlations needed to reach the Heisenberg scaling of the quantum Fisher information—a quadratic improvement in precision with atom number—in a stationary, dissipative state, with no direct atom-atom interactions. The argument contrasts two models: a semiclassical one that treats atoms as independent and can violate the quantum Cramér-Rao bound by orders of magnitude for large N and weak dissipation, and a collective model that respects the bound. Because the collective model contains no interaction terms, the correlations responsible for restoring the bound and producing Heisenberg scaling must be induced by the measurement itself. A reader should care because this identifies measurement-induced correlations as a quantum resource for sensing, suggests that existing alkali-vapor magnetometers may operate on collective quantum effects, and yields an experimentally testable consistency condition for macroscopic quantum mechanics.

What carries the argument

The key machinery is a generalized master equation with full-counting statistics, derived from a quantum-trajectory description of the light-matter interaction, which guarantees microscopic consistency with the quantum Cramér-Rao bound. For the collective model, the paper analyzes the equation in the thermodynamic limit via a non-unitary mean-field transformation based on the Holstein-Primakoff representation of the collective spin. The load-bearing identity is Eq. (29), the analytic expression for the quantum Fisher information, which separates an N-linear pumping term (κ_P) from an N-quadratic collective-decay term (κ_z); the latter, when κ_z ∝ N, is the source of Heisenberg scaling. The m

What would settle it

Measure the quantum Fisher information (or the signal-to-noise ratio) of an optical magnetometer as a function of atom number N above 10^10. If it scales linearly in N, the predicted Heisenberg scaling is falsified. Alternatively, within the paper's own framework, re-derive Eq. (29) with the appendix's scaling γ_z ∝ 1/N: the N² term disappears, so the presence or absence of a genuine quadratic term can be checked analytically.

Watch

Extended reading notes

Core claim

The paper's central claim is that the quantum Fisher information of a collective optical magnetometer, evaluated analytically in the stationary state, contains a term quadratic in the atom number N (Eq. (29)). With κ_z = N γ_z Ω²/ε², the term N κ_z gives an N² contribution, which is the Heisenberg scaling. The authors attribute this to measurement-induced correlations: the model contains no inter-atomic interaction terms, yet the semiclassical independent-atom model violates the quantum Cramér-Rao bound while the collective model respects it. The bound-violation and the presence of the N² term together identify the measurement-generated correlations as the resource that restores the bound an

Load-bearing premise

The quadratic (Heisenberg) scaling of the quantum Fisher information rests on the assumption that the collective decay rate γ_z—the rate of spontaneous emission into the probe beam direction—is independent of the atom number N; the manuscript's own appendix assumes instead that γ_z ∝ 1/N, which removes the quadratic term.

Editorial extensions

If this is right

  • If the prediction holds, optical magnetometers with large atomic ensembles (N ≳ 10^10) could reach the Heisenberg limit without any engineered squeezing or entanglement, purely from the measurement dynamics.
  • The quantum Cramér-Rao bound becomes a practical diagnostic: comparing measured signal-to-noise ratios with the predicted Fisher information can reveal whether a semiclassical description is invalid and collective correlations are present.
  • The linear-in-time dependence of the quantum Fisher information in Eq. (29) contradicts earlier τ³ predictions from Gaussian approximations, so measurement duration enters the precision scaling differently than previously assumed.
  • The work provides a concrete route to test the foundations of quantum mechanics on macroscopic ensembles: if the measured SNR and the Fisher information violate the consistency condition of Eq. (30), it would signal either missing collective correlations or a problem with the quantum description of light-matter interaction.

Reading between the lines

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

  • An experiment that directly measures the scaling exponent of the quantum Fisher information (or the SNR) with atom number in an optical magnetometer could discriminate between the collective model (quadratic) and the semiclassical model (linear); such a measurement would also resolve the ambiguity in how γ_z scales with N.
  • The paper's own appendix shows that if γ_z must be rescaled as 1/N to obtain a well-defined thermodynamic limit, the quadratic term in Eq. (29) vanishes, leaving only linear scaling. Whether the physical γ_z is fixed or N-dependent is thus the decisive open question the authors do not settle.
  • The idea that measurement-induced correlations are a quantum resource may generalize to other continuous-measurement sensors (e.g., atomic clocks, electrometry), suggesting that dissipative steady states can host quantum-enhanced sensitivity without interaction engineering.
  • The connection to spin squeezing via measurement (as in cavity QED) but in free space suggests that the mechanism may be a general feature of indistinguishable-emitter ensembles under continuous observation, with potential applications in distributed sensing.
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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

3 major / 4 minor

Summary. The paper analyzes optical magnetometry with two models. The first is a semiclassical, independent-atom model; the second is a collective-spin model in which all atoms share the same light-matter coupling. The authors claim that the semiclassical model can violate the quantum Cramér-Rao bound by orders of magnitude in the large-N, weak-dissipation regime, and that the collective model respects the bound while predicting Heisenberg (N²) scaling of the quantum Fisher information in a dissipative steady state, with correlations generated by the measurement itself. The central quantitative formula is Eq. (29), which contains terms quadratic in N when the collective decay parameter κ_z is taken ∝N. The paper also develops a full-counting-statistics quantum-trajectory framework and provides finite-size benchmarks.

Significance. If the Heisenberg-scaling claim were sound, the paper would identify a new resource for quantum-enhanced sensing and a new paradigm for open-system metrology. The manuscript has several genuine strengths: the full-counting-statistics framework is a useful methodological contribution; the use of the quantum Cramér-Rao bound as a sanity check for semiclassical spectroscopic theories is conceptually valuable; and the authors provide numerical data and source-code links. However, the central claim is not consistently derived. The scaling of γ_z with N is treated differently in the main text and in the appendices, and where the two treatments conflict, the paper's own benchmarks show that the mean-field method used to obtain Eq. (29) breaks down exactly in the regime that would give Heisenberg scaling. As presented, the paper does not establish measurement-induced Heisenberg scaling.

major comments (3)
  1. [§III B, Eq. (29), and Appendix E2] The Heisenberg scaling in Eq. (29) is an artifact of the assumed N-dependence of γ_z. The N² terms appear because κ_z = Nγ_zΩ²/ε_Δ² is treated as ∝ N, i.e., γ_z is held fixed as N grows. However, Appendix E2 states that a well-defined thermodynamic limit requires 'γ_P, γ_D and γ_z scale as N^{-1}'. Under that scaling κ_z is constant and Eq. (29) gives only linear scaling. The paper never states which scaling is physical, nor why the fixed-γ_z case can be analyzed with a mean-field expansion whose derivation assumes γ_z∼1/N. Since the title and abstract rest on this N² term, this is a load-bearing inconsistency.
  2. [Appendix E4, Fig. 5] Eq. (29) is derived from the non-unitary mean-field theory of Appendix E2, which assumes a spin coherent state on the Bloch sphere (α=0, i.e., ⟨S_x⟩/N=-1/2). The finite-size benchmarks in Fig. 5 show that for κ_z≳0.1h_x — precisely the regime in which Eq. (29) produces the N² scaling — the exact stationary state instead has ⟨S_x⟩=0, i.e., the spin lies at the center of the Bloch sphere. The authors explicitly state that the mean-field approach 'fails to agree with finite-size calculations in this atom-number regime.' Thus Eq. (29) is not a valid thermodynamic-limit expression in the regime where it predicts the headline effect.
  3. [Appendix E4, Eq. (E32)] The finite-size master equation used for benchmarking parameterizes the collective dissipator as (κ_z/N)∑D[P_a], i.e., γ_z∝1/N. This contradicts the fixed-γ_z choice in Sec. III B that generates κ_z∝N. The paper invokes two incompatible scalings without explaining when each applies; the benchmark model, which is the one said to represent the well-defined thermodynamic limit, never produces the N² term. The manuscript therefore lacks a single consistent scaling hypothesis under which the central claim survives.
minor comments (4)
  1. [General] Typos and wording: 'direclty' in Sec. III C; 'Cramer-Round inequality' in the model-limitations paragraph; 'thermodynamics limit' in Appendix E2. Please copy-edit.
  2. [Eq. (29) vs Eq. (E33)] The definition of κ_z differs between the main text (κ_z = Nγ_zΩ²/ε_Δ²) and Appendix E4 (κ_z = Nγ_zΩ²/(ε_Δ²+γ_z²/4)). This notational inconsistency should be fixed.
  3. [Sec. III B] The paragraph introducing κ_P and κ_z says a rescaling 'is not feasible for the dissipation resulting from the excited-states' but does not explicitly state how γ_z is treated in the collective-model calculations. Please state the scaling of γ_z with N explicitly at that point.
  4. [Fig. 5 and surrounding text] The crossover value is described as 'κ_z ≈0.1h_x' in two places; clarify whether the text means κ_z ≥ 0.1h_x and give the value in the caption.

Circularity Check

1 steps flagged · score 8.0 of 10

Heisenberg scaling in Eq. (29) is forced by the choice to keep γ_z fixed; the paper's own thermodynamic-limit derivation assumes γ_z ∝ 1/N, making the N² term vanish.

  1. self definitional [Eq. (29); Sec. III B; App. E 2; App. E 4]
    "where h_x = µB_x, and κ_z = N γ_z Ω^2/ε_Δ^2 ... Now I_Bz^(Q) increases quadratically, i.e., it exhibits Heisenberg scaling. From this scaling, we conclude that the Fisher information leaks via the collective dissipation ∝ γ_z N ... For the generalized master equation in Eq. (E11), we formally assume that γ_P, γ_D and γ_z scale as N^{-1} to obtain a well-defined thermodynamic limit."

    The N² term in Eq. (29) is proportional to κ_z = N γ_z Ω²/ε_Δ². The paper obtains the quadratic term by keeping γ_z fixed, so κ_z ∝ N. But the paper's own mean-field derivation in App. E2 is formulated for γ_z ∼ N^{-1}, under which κ_z is constant and Eq. (29) predicts only linear scaling; App. E4 benchmarks with κ_z/N, i.e. constant κ_z. Thus the 'Heisenberg scaling' statement is a restatement of the γ_z-scaling choice, not a consequence of measurement-induced correlations.

full rationale

The central claim — measurement-induced Heisenberg scaling of the quantum Fisher information — is not an independent prediction of the model. In Eq. (29), the term quadratic in N arises from κ_z = N γ_z Ω²/ε_Δ² only when γ_z is held fixed while N grows (Sec. III B). The paper's own non-unitary mean-field method in Appendix E2 is explicitly derived under the assumption γ_z ∼ 1/N, which makes κ_z constant and reduces Eq. (29) to linear (standard quantum limit) scaling. Appendix E4 likewise parameterizes the master equation with κ_z/N, a constant κ_z in the thermodynamic limit, and its finite-size benchmarks exhibit only the linear-in-κ_z regime (with a √8 pre-convergence discrepancy). The manuscript never reconciles the fixed-γ_z assumption used for the headline scaling with the γ_z ∼ 1/N assumption needed for the derivation of Eq. (29) and for the benchmarks. In the fixed-γ_z regime, the collective dissipators grow faster than N and the mean-field expansion of App. E2 is not valid; in the γ_z ∼ 1/N regime, the Heisenberg scaling disappears. Therefore the claimed N² scaling reduces to the chosen normalization of γ_z rather than following from the physics of measurement-induced correlations. No self-citation chain or imported uniqueness theorem is load-bearing here; the circularity is definitional in the scaling of the dissipation rate.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new particles, fields, or forces. The central claim rests on modeling choices: a phenomenological pump rate, the N-scaling of the collective decay rate γ_z, and the all-atoms-at-one-position idealization of the collective model. The most fragile item is the γ_z scaling, which determines whether the predicted QFI is quadratic or linear in N.

free parameters (3)
  • γ_P (optical pumping rate) = 30 kHz in numerical examples
    Phenomenological pump term introduced in Eqs. (10) and (23); not derived microscopically. Controls the CRB-violation ratio µB_x/γ_P and appears in the QFI prefactor of Eq. (29).
  • N-scaling of γ_z (collective decay along probe direction) = main text: N^0; Appendix E2: N^{-1}
    The N² Heisenberg term in Eq. (29) is proportional to κ_z=Nγ_zΩ²/ε², so it exists only if γ_z is kept fixed. Appendix E2 states a well-defined thermodynamic limit requires γ_z∼N^{-1}, under which κ_z is constant and QFI is linear.
  • Bias field h_x = µB_x = µB_x/ℏ = 500 kHz
    Operating point chosen from experiment; appears in the denominator of Eq. (29) and in the semiclassical CRB-violation criterion µB_x>γ_P. Not fitted to the target result.
assumptions (6)
  • standard math Markovian quantum-trajectory master equation with full-counting statistics
    Appendix B derives the generalized master equation from a coherent pulse-train light-matter interaction under Born-Markov assumptions. This is a standard framework used throughout the paper.
  • domain assumption Large-detuning adiabatic elimination of the excited-state manifold
    Both models project onto the ground-state manifold assuming ϵΔ≫Γ (Appendices D2, E1). This is standard but restricts validity to the off-resonant regime relevant for optical magnetometry.
  • domain assumption Semiclassical model: independent atoms and Gaussian stochastic photonic mean field
    Appendix C assumes independent atoms and a Gaussian distribution of the photonic mean field, dropping atom-atom correlations. The paper itself uses this assumption to expose the CRB violation.
  • domain assumption Collective model: all atoms at the same position r_m=r_0 and no local dissipation (γ_D=0)
    Sec. III B assumes all atoms are indistinguishable with respect to the light-matter interaction. The paper acknowledges this 'strictly speaking does not reflect the experimental circumstances' and that local spontaneous emission would hurt collective correlations.
  • ad hoc to paper Well-defined thermodynamic limit requires γ_P, γ_D and γ_z to scale as N^{-1}
    Appendix E 2 states this formally to obtain a well-defined N→∞ limit, but the main text (Sec. III B) says γ_z cannot be rescaled because it is fixed by Eq. (B20). The contradiction is load-bearing for the Heisenberg-scaling claim.
  • standard math Non-unitary Holstein-Primakoff mean-field theory to leading order in N
    Appendix E 2 uses the Holstein-Primakoff transformation and a four-component non-unitary shift to evaluate cumulants in the thermodynamic limit. The paper's own Fig. 5 shows this approach becomes invalid or only 'pre-converged' for large κ_z.

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Pith. "Pith review of Heisenberg scaling in optical magnetometry with measurement-induced correlations as a quantum resource." pith.science (2026). https://pith.science/paper/FBZFCCBU

@misc{pith2026260101820,
  author       = {Pith},
  title        = {Pith review of: Heisenberg scaling in optical magnetometry with measurement-induced correlations as a quantum resource},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FBZFCCBU}},
  note         = {Machine review of arXiv:2601.01820}
}
read the original abstract

Theoretical proposals to reach the Heisenberg scaling of the measurement precision typically require carefully engineered interactions or initial entanglement. In studying optical magnetometry, we show that the continuous collective measurement process itself can generate the necessary many-body quantum correlations to achieve the elusive Heisenberg scaling of the quantum Fisher information in a dissipative, steady-state system without direct inter-atomic interactions. By contrasting a correlation-neglecting but otherwise consistent semiclassical model, which can violate the quantum Cram\'er-Rao bound (QCRB) by several orders of magnitude, with a collective quantum model, we isolate measurement-induced correlations as the essential mechanism. The violation of the QCRB serves thereby as a fundamental sanity check for semiclassical spectroscopic theories. This work reveals measurement-induced correlations as a widely unexplored quantum resource for quantum-enhanced sensing, establishes a new paradigm for achieving Heisenberg scaling in open quantum systems, and provides a direct path to test the foundations of quantum mechanics using macroscopic atom ensembles.

Figures

Figures reproduced from arXiv: 2601.01820 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Sketch of the optical magnetometer, which utilizes the rotation of linearly polarized laser light induced by the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Investigation of the semiclassical model. (a) Expectation value of the spin operators as a function of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Investigation of the collective model. (a) Expectation value of the spin operators as a function of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Benchmark calculations of the collective model in [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Expectation value of [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]

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Works this paper leans on

107 extracted references · 1 canonical work pages

  1. [1]

    (6) for Hamiltonian parameter estimation using an argument from linear-response theory

    Modified Cramer-Rao bound Here, we derive the representation of the Fisher infor- mation in Eq. (6) for Hamiltonian parameter estimation using an argument from linear-response theory. For a more general derivation, we refer to Ref. [80]. Thereby, we assume that the Hamiltonian depends on the param- eter to be measuredX, i.e., ˆH= ˆHX . Applying the Heisen...

  2. [2]

    (5) which are easier to evaluate

    Quantum Fisher information Here we provide alternative expressions for the quan- tum Fisher information in Eq. (5) which are easier to evaluate. The time-evolved wave function at timeτreads |ΨX ⟩=e −i ˆHX τ |Ψ(0)⟩,(A6) where|Ψ(0)⟩denotes the initial state. We now show that the Fisher information can be expressed as [81] I (Q) X = 4 h ⟨∂X ΨX |∂ X ΨX ⟩ − |⟨...

  3. [3]

    F ull-counting statistics We are interested into the measurement statistics of a set of photonic occupation operators ˆn ξ = ˆa† ξˆaξ, which will be specified in Sec. B 2. The corresponding photon numbers are denoted byn ξ. In terms of the photonic probabilitiesp n, wheren= (n 1, . . . , nξmax ) is a vector of photon numbers, the moment- and cumulant gene...

  4. [4]

    For the semiclas- sical model, the matter system is a single atom

    Generalized master equation In this Appendix, we derive an expression for the moment-generating function of the photonic field after interaction with the matter system located at a partic- ular positionrwhich is on an equal footing with the quantum trajectory approach [66, 67]. For the semiclas- sical model, the matter system is a single atom. For the col...

  5. [5]

    As explained in Eq

    Time-integrated polarization measurements In the previous derivation, we have assumed a generic time-resolved measurement, which has attributed to each time increment its own counting fieldχ ξ withξ= (η, tξ), whereη∈ {x,y}denotes the polarization direc- tion. As explained in Eq. (7), we are interested into time-integrated measurements, which distinquish o...

  6. [6]

    Here, we show how the cumulants can be expressed in an integral form as given in the main text

    Integral expression of the cumulants In Appendix B 2, we explained how to calculate the cumulants of the photonic probability distribution using the generalized master equation. Here, we show how the cumulants can be expressed in an integral form as given in the main text. For simplicity, we restrict the explanations to a single counting fieldχ. To this e...

  7. [7]

    (B6) with polarization di- rectionη∈ {x,y}

    Semiclassical flow equations In the semiclassical framework, we considerstochastic photonic mean fieldsν η =|α η|2, whereα η describes the coherent photonic state in Eq. (B6) with polarization di- rectionη∈ {x,y}. The following treatment focuses on the first- and second-order cumulants, which we investigate in this work. The stochastic mean fields are des...

  8. [8]

    Statistics of polarization measurements The measurement statistics of the polarization rotation in a magnetometer can be described using two photonic modes representing the two polarization directions of the linearly polarized light [i.e., the x and y directions in Fig. 1(a)]. We denote the photon-number operators by ˆnx and ˆny. The related stochastic me...

Show all 107 references
  1. [9]

    Generalized master equation Following the procedure in Appendix B 2, we construct the generalized master equation in Eq. (B16). In doing so, the semiclassical Hamiltonian corresponding to the microscopic Hamiltonian in Eq. (2) is given by ˆHm,χ(t) = ˆHM,m + X a=± Ωa,0 2 |e−a⟩ ...

  2. [10]

    Effective two-level system To enable an analytical investigation, we derive here an effective two-level system, which inherits the essential physical properties of the four-level system in Eq. (D1). To this end, we first construct the equations of motions of the density matrix...

  3. [11]

    (D5), we take advantage 20 of an expansion method for the characteristic polynomial of the Liouvillian in Eq

    Analytical calculations To efficiently calculate the first two cumulants in the long-time limit according to Eq. (D5), we take advantage 20 of an expansion method for the characteristic polynomial of the Liouvillian in Eq. (B16) [84]. The characteristic polynomial can be expan...

  4. [12]

    Master equation Using the formalism in Appendix B 2, we can establish a generalized quantum-master equation, which reads d dt ρχ =−i h ˆHL(t)ρχ −ρ χ ˆHR(t) i +γ z X a=± X η=x,y Dχη [ ˆJ − a,−a]ρχ + 2γ D X a=± D[ ˆJ − a,−a]ρχ +γ D X a=± D[ ˆJ − a,a]ρχ.(E4) Thereby, we have intr...

  5. [13]

    Non-unitary mean-field theory As the ensemble consists of a macroscopic number of atoms, we carry out a mean-field treatment to analyze the measurement statistics. As a first step, we apply the Holstein-Primakoff transformation ˆSx = ˆa†ˆa−N 2 , ˆS(x) + = ˆSz −i ˆSy = ˆa† p N−...

  6. [14]

    Crucially, the mean-fields appearing in Eq

    Efficient evaluation of low-order cumulants In this Appendix, we explain how to efficiently calcu- late low-order cumulants of the measurement statistics. Crucially, the mean-fields appearing in Eq. (E11) are de- 24 termined as the roots ofl f , lf∗ , lb, lb∗ in Eq. (E18), and...

  7. [15]

    For this reason, it is possible to evaluate the quantum Fisher information analytically, which give the result in Eq. (29)

  8. [16]

    To achieve a well- defined thermodynamic limit and for the simplicity of notation, we parameterize the generalized Liouvillian in Eq

    Benchmark calculations To validate the non-Hermitian mean-field theory, we compare the mean-field approach with exact numerical calculations of the finite-size model. To achieve a well- defined thermodynamic limit and for the simplicity of notation, we parameterize the general...

  9. [17]

    Ye and P

    J. Ye and P. Zoller, Essay: Quantum sensing with atomic, molecular, and optical platforms for fundamental physics, Phys. Rev. Lett.132, 190001 (2024)

    2024

  10. [18]

    C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys.89, 035002 (2017)

    2017

  11. [19]

    A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, Optical atomic clocks, Rev. Mod. Phys.87, 637 (2015)

    2015

  12. [20]

    G. Hoth, A. Radnaev, P. Mitchell, J. Sherman, and V. Gerginov, ”NIST-F3, a cesium fountain frequency ref- erence”, inProceedings of the 54th Annual Precise Time and Time Interval Systems and Applications (PTTI) Meeting(2024) pp. 183–188

    2024

  13. [21]

    S. M. Brewer, J.-S. Chen, A. M. Hankin, E. R. Clements, C. W. Chou, D. J. Wineland, D. B. Hume, and D. R. Leibrandt, 27Al+ Quantum-Logic Clock with a Sys- tematic Uncertainty below 10 −18, Phys. Rev. Lett.123, 033201 (2019)

    2019

  14. [22]

    N. Ji, Y. Liang, W. Yu, Q. Yin, and T. Vogt, Optimiz- ing four-wave mixing in Rydberg atoms for microwave- optical conversion, Phys. Rev. A110, 052608 (2024)

    2024

  15. [23]

    D. H. Meyer, Z. A. Castillo, K. C. Cox, and P. D. Kunz, Assessment of Rydberg atoms for wideband electric field sensing, Journal of Physics B: Atomic, Molecular and Optical Physics53, 034001 (2020)

    2020

  16. [24]

    M. Jing, Y. Hu, J. Ma, H. Zhang, L. Zhang, L. Xiao, and S. Jia, Atomic superheterodyne receiver based on microwave-dressed Rydberg spectroscopy, Nature Physics16, 911 (2020)

    2020

  17. [25]

    D. Li, Z. Bai, X. Zuo, Y. Wu, J. Sheng, and H. Wu, Room temperature single-photon terahertz detection with ther- mal rydberg atoms, Applied Physics Reviews11, 041420 (2024)

    2024

  18. [26]

    Engelhardt, K

    G. Engelhardt, K. Dorfman, and Z. Zhang, Photon- resolved floquet theory approach to spectroscopic quan- tum sensing, Phys. Rev. A112, 063725 (2025)

    2025

  19. [27]

    K. E. Dorfman, S. Asban, B. Gu, and S. Mukamel, Hong- Ou-Mandel interferometry and spectroscopy using entan- gled photons, Communications Physics4, 49 (2021)

    2021

  20. [28]

    Zhang, T

    Z. Zhang, T. Peng, X. Nie, G. S. Agarwal, and M. O. Scully, Entangled photons enabled time-frequency- resolved coherent Raman spectroscopy and applications to electronic coherences at femtosecond scale, Light: Sci- ence and Applications11, 10.1038/s41377-022-00953-y (2022)

    2022 doi

  21. [29]

    T. Li, F. Li, X. Liu, V. V. Yakovlev, and G. S. Agar- wal, Quantum-enhanced stimulated Brillouin scattering spectroscopy and imaging, Optica9, 959 (2022)

    2022

  22. [30]

    Zhang, X

    Z. Zhang, X. Zhang, J. Liu, and H. Dong, Quantum- enhanced weak absorption estimation with correlated photons, Phys. Rev. Lett.134, 133604 (2025)

    2025

  23. [31]

    A. L. Bloom, Principles of operation of the rubidium va- pormagnetometer, Appl. Opt.1, 61 (1962)

    1962

  24. [32]

    J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Ro- malis, High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation, Phys. Rev. Lett.89, 130801 (2002)

    2002

  25. [33]

    V. Shah, S. Knappe, P. D. D. Schwindt, and J. Kitch- 27 ing, Subpicotesla atomic magnetometry with a microfab- ricated vapour cell, Nature Photonics1, 649 (2007)

    2007

  26. [34]

    Wolfgramm, A

    F. Wolfgramm, A. Cer` e, F. A. Beduini, A. Predojevi´ c, M. Koschorreck, and M. W. Mitchell, Squeezed-light op- tical magnetometry, Phys. Rev. Lett.105, 053601 (2010)

    2010

  27. [35]

    Budker and D

    D. Budker and D. F. Jackson Kimball, eds.,Optical Mag- netometry(Cambridge University Press, 2013)

    2013

  28. [36]

    Jim´ enez-Mart ´ ınez, J

    R. Jim´ enez-Mart ´ ınez, J. Ko lody´ nski, C. Troullinou, V. G. Lucivero, J. Kong, and M. W. Mitchell, Signal tracking beyond the time resolution of an atomic sensor by kalman filtering, Phys. Rev. Lett.120, 040503 (2018)

    2018

  29. [37]

    Cohen, K

    Y. Cohen, K. Jadeja, S. Sula, M. Venturelli, C. Deans, L. Marmugi, and F. Renzoni, A cold atom radio- frequency magnetometer, Applied Physics Letters114, 073505 (2019)

    2019

  30. [38]

    Y. Ma, Y. Chen, M. Yu, Y. Wang, S. Lu, J. Guo, G. Luo, L. Zhao, P. Yang, Q. Lin, and Z. Jiang, Ultrasensitive serf atomic magnetometer with a miniaturized hybrid vapor cell, Microsystems and Nanoengineering10, 121 (2024)

    2024

  31. [39]

    H. Bao, J. Duan, S. Jin, X. Lu, P. Li, W. Qu, M. Wang, I. Novikova, E. E. Mikhailov, K.-F. Zhao, K. Mølmer, H. Shen, and Y. Xiao, Spin squeezing of 10 11 atoms by prediction and retrodiction measurements, Nature581, 159 (2020)

    2020

  32. [40]

    S. Jin, J. Duan, Y. Zhang, X. Zhang, H. Bao, H. Shen, L. Xiao, S. Jia, M. Wang, and Y. Xiao, Concurrent spin squeezing and light squeezing in an atomic ensemble, Phys. Rev. Lett.133, 173604 (2024)

    2024

  33. [41]

    Babaei, B

    B. Babaei, B. D. Smith, A. Tretiakov, A. Narayanan, and L. J. LeBlanc, Microwave-optical double-resonance vector magnetometry with warm rb atoms (2025), arXiv:2507.08791 [physics.atom-ph]

    2025 arXiv

  34. [42]

    Hammerer, A

    K. Hammerer, A. S. Sørensen, and E. S. Polzik, Quantum interface between light and atomic ensembles, Rev. Mod. Phys.82, 1041 (2010)

    2010

  35. [43]

    B. Chen, C. Qiu, S. Chen, J. Guo, L. Q. Chen, Z. Y. Ou, and W. Zhang, Atom-light hybrid interferometer, Phys. Rev. Lett.115, 043602 (2015)

    2015

  36. [44]

    X. Wang, J. Wang, Z. Ren, R. Wen, C.-L. Zou, G. A. Siviloglou, and J. F. Chen, Quantum interference be- tween photons and single quanta of stored atomic co- herence, Phys. Rev. Lett.128, 083605 (2022)

    2022

  37. [45]

    I. M. Bloch, Y. Hochberg, E. Kuflik, and T. Volansky, Axion-like relics: new constraints from old comagne- tometer data, Journal of High Energy Physics2020, 167 (2020)

    2020

  38. [46]

    Afach, B

    S. Afach, B. C. Buchler, D. Budker, C. Dailey, A. Dere- vianko, V. Dumont, N. L. Figueroa, I. Gerhardt, Z. D. Gruji´ c, H. Guo, C. Hao, P. S. Hamilton, M. Hedges, D. F. Jackson Kimball, D. Kim, S. Khamis, T. Kor- nack, V. Lebedev, Z.-T. Lu, H. Masia-Roig, M. Monroy, M. Padniuk...

    2021

  39. [47]

    I. M. Bloch, G. Ronen, R. Shaham, O. Katz, T. Volansky, and O. Katz, New constraints on axion-like dark matter using a Floquet quantum detector, Science Advances8, eabl8919 (2022)

    2022

  40. [48]

    C. B. Møller, R. A. Thomas, G. Vasilakis, E. Zeuthen, Y. Tsaturyan, M. Balabas, K. Jensen, A. Schliesser, K. Hammerer, and E. S. Polzik, Quantum back-action- evading measurement of motion in a negative mass ref- erence frame, Nature547, 191 (2017)

    2017

  41. [49]

    F. Y. Khalili and E. S. Polzik, Overcoming the standard quantum limit in gravitational wave detectors using spin systems with a negative effective mass, Phys. Rev. Lett. 121, 031101 (2018)

    2018

  42. [50]

    J. Jia, V. Novikov, T. B. Brasil, E. Zeuthen, J. H. M¨ uller, and E. S. Polzik, Acoustic frequency atomic spin oscilla- tor in the quantum regime, Nature Communications14, 6396 (2023)

    2023

  43. [51]

    Novikov, J

    V. Novikov, J. Jia, T. B. Brasil, A. Grimaldi, M. Bo- coum, M. Balabas, J. H. M¨ uller, E. Zeuthen, and E. S. Polzik, Hybrid quantum network for sensing in the acous- tic frequency range, Nature643, 955 (2025)

    2025

  44. [52]

    Cram´ er,Mathematical Methods of Statistics(Prince- ton University Press, Princeton, 1946)

    H. Cram´ er,Mathematical Methods of Statistics(Prince- ton University Press, Princeton, 1946)

    1946

  45. [53]

    C. R. Rao, Information and accuracy attanaible in the estimation of statistical parameters, Bull. Calcutta Math. Soc.37, 81 (1945)

    1945

  46. [54]

    Helstrom, Minimum mean-squared error of estimates in quantum statistics, Physics Letters A25, 101 (1967)

    C. Helstrom, Minimum mean-squared error of estimates in quantum statistics, Physics Letters A25, 101 (1967)

    1967

  47. [55]

    Hayashi,Quantum Information Theory(Springer Berlin, Heidelberg, 2017)

    M. Hayashi,Quantum Information Theory(Springer Berlin, Heidelberg, 2017)

    2017

  48. [56]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Quantum- enhanced measurements: Beating the standard quantum limit, Science306, 1330 (2004)

    2004

  49. [57]

    R. Liu, Y. Chen, M. Jiang, X. Yang, Z. Wu, Y. Li, H. Yuan, X. Peng, and J. Du, Experimental critical quan- tum metrology with the Heisenberg scaling, npj Quan- tum Information7, 170 (2021)

    2021

  50. [58]

    Di Candia, F

    R. Di Candia, F. Minganti, K. V. Petrovnin, G. S. Paraoanu, and S. Felicetti, Critical parametric quantum sensing, npj Quantum Information9, 23 (2023)

    2023

  51. [59]

    L. Qin, J. Li, Y. Niu, and X.-Q. Li, Enhanced super- Heisenberg scaling precision by nonlinear coupling and postselection, Physics Letters A523, 129795 (2024)

    2024

  52. [60]

    Block, B

    M. Block, B. Ye, B. Roberts, S. Chern, W. Wu, Z. Wang, L. Pollet, E. J. Davis, B. I. Halperin, and N. Y. Yao, Scal- able spin squeezing from finite-temperature easy-plane magnetism, Nature Physics20, 1575 (2024)

    2024

  53. [61]

    Sarkar, A

    S. Sarkar, A. Bayat, S. Bose, and R. Ghosh, Exponentially-enhanced quantum sensing with many- body phase transitions, Nature Communications16, 5159 (2025)

    2025

  54. [62]

    Malnou, D

    M. Malnou, D. A. Palken, B. M. Brubaker, L. R. Vale, G. C. Hilton, and K. W. Lehnert, Squeezed vacuum used to accelerate the search for a weak classical signal, PRX 9, 021023 (2019)

    2019

  55. [63]

    Gessner, A

    M. Gessner, A. Smerzi, and L. Pezz` e, Multiparameter squeezing for optimal quantum enhancements in sensor networks, Nature Communications11, 3817 (2020)

    2020

  56. [64]

    K. M. Backes, D. A. Palken, S. A. Kenany, B. M. Brubaker, S. B. Cahn, A. Droster, G. C. Hilton, S. Ghosh, H. Jackson, S. K. Lamoreaux, A. F. Leder, K. W. Lehnert, S. M. Lewis, M. Malnou, R. H. Maruyama, N. M. Rapidis, M. Simanovskaia, S. Singh, D. H. Speller, I. Urdinaran, L. ...

    2021

  57. [65]

    T.-W. Mao, Q. Liu, X.-W. Li, J.-H. Cao, F. Chen, W.- 28 X. Xu, M. K. Tey, Y.-X. Huang, and L. You, Quantum- enhanced sensing by echoing spin-nematic squeezing in atomic Bose-Einstein condensate, Nature Physics19, 1585 (2023)

    2023

  58. [66]

    E. M. Kessler, P. K´ om´ ar, M. Bishof, L. Jiang, A. S. Sørensen, J. Ye, and M. D. Lukin, Heisenberg-limited atom clocks based on entangled qubits, Phys. Rev. Lett. 112, 190403 (2014)

    2014

  59. [67]

    K´ om´ ar, E

    P. K´ om´ ar, E. M. Kessler, M. Bishof, L. Jiang, A. S. Sørensen, J. Ye, and M. D. Lukin, A quantum network of clocks, Nature Physics10, 582 (2014)

    2014

  60. [68]

    Demkowicz-Dobrza´ nski and L

    R. Demkowicz-Dobrza´ nski and L. Maccone, Using entan- glement against noise in quantum metrology, Phys. Rev. Lett.113, 250801 (2014)

    2014

  61. [69]

    Demkowicz-Dobrza´ nski, J

    R. Demkowicz-Dobrza´ nski, J. Czajkowski, and P. Sekatski, Adaptive quantum metrology under general Markovian noise, Phys. Rev. X7, 041009 (2017)

    2017

  62. [70]

    S. Zhou, M. Zhang, J. Preskill, and L. Jiang, Achiev- ing the Heisenberg limit in quantum metrology using quantum error correction, Nature Communications9, 78 (2018)

    2018

  63. [71]

    Didier, F

    N. Didier, F. Qassemi, and A. Blais, Perfect squeezing by damping modulation in circuit quantum electrody- namics, Phys. Rev. A89, 013820 (2014)

    2014

  64. [72]

    Blais, A

    A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys.93, 025005 (2021)

    2021

  65. [73]

    Ilias, D

    T. Ilias, D. Yang, S. F. Huelga, and M. B. Plenio, Criticality-enhanced quantum sensing via continuous measurement, PRX Quantum3, 010354 (2022)

    2022

  66. [74]

    Cabot, F

    A. Cabot, F. Carollo, and I. Lesanovsky, Continuous sensing and parameter estimation with the boundary time crystal, Phys. Rev. Lett.132, 050801 (2024)

    2024

  67. [75]

    W. Qin, A. Miranowicz, and F. Nori, Exponentially improved dispersive qubit readout with squeezed light, Phys. Rev. Lett.133, 233605 (2024)

    2024

  68. [76]

    L. K. Thomsen, S. Mancini, and H. M. Wiseman, Spin squeezing via quantum feedback, Phys. Rev. A65, 061801 (2002)

    2002

  69. [77]

    Geremia, J

    J. Geremia, J. K. Stockton, A. C. Doherty, and H. Mabuchi, Quantum kalman filtering and the heisen- berg limit in atomic magnetometry, Phys. Rev. Lett.91, 250801 (2003)

    2003

  70. [78]

    Mølmer and L

    K. Mølmer and L. B. Madsen, Estimation of a classical parameter with Gaussian probes: Magnetometry with collective atomic spins, Phys. Rev. A70, 052102 (2004)

    2004

  71. [79]

    L. B. Madsen and K. Mølmer, Spin squeezing and pre- cision probing with light and samples of atoms in the gaussian description, Phys. Rev. A70, 052324 (2004)

    2004

  72. [80]

    Zhang and K

    C. Zhang and K. Mølmer, Estimating a fluctuating mag- netic field with a continuously monitored atomic ensem- ble, Phys. Rev. A102, 063716 (2020)

    2020

  73. [81]

    H. M. Wiseman and G. J. Milburn, Quantum theory of field-quadrature measurements, Phys. Rev. A47, 642 (1993)

    1993

  74. [82]

    Wiseman and G

    H. Wiseman and G. Milburn,Quantum measurement and control(Cambridge University Press, New York, 2010)

    2010

  75. [83]

    G. T. Landi, M. J. Kewming, M. T. Mitchison, and P. P. Potts, Current fluctuations in open quantum sys- tems: Bridging the gap between quantum continuous measurements and full counting statistics, PRX Quan- tum5, 020201 (2024)

    2024

  76. [84]

    Sch¨ on, E

    C. Sch¨ on, E. Solano, F. Verstraete, J. I. Cirac, and M. M. Wolf, Sequential generation of entangled multi- qubit states, Phys. Rev. Lett.95, 110503 (2005)

    2005

  77. [85]

    J. I. Cirac, D. P´ erez-Garc ´ ıa, N. Schuch, and F. Ver- straete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys.93, 045003 (2021)

    2021

  78. [86]

    D. Yang, S. F. Huelga, and M. B. Plenio, Efficient infor- mation retrieval for sensing via continuous measurement, Phys. Rev. X13, 031012 (2023)

    2023

  79. [87]

    Mukamel,Principles of nonlinear optical spectroscopy (OXFORD UNIVERSITY PRESS, 1995)

    S. Mukamel,Principles of nonlinear optical spectroscopy (OXFORD UNIVERSITY PRESS, 1995)

    1995

  80. [88]

    M. O. Scully and M. S. Zubairy,Quantum optics(Cam- bridge Unversity Press, 1997)

    1997

  81. [89]

    Pustelny, W

    S. Pustelny, W. Gawlik, S. M. Rochester, D. F. J. Kim- ball, V. V. Yashchuk, and D. Budker, Nonlinear magneto- optical rotation with modulated light in tilted magnetic fields, Phys. Rev. A74, 063420 (2006)

    2006

  82. [90]

    Auzinsh, D

    M. Auzinsh, D. Budker, and S. Rochester,Optically Polarized Atoms: Understanding light-atom interactions (Oxford University Press, 2010)

    2010

  83. [91]

    A. e. a. Akbar, Optimized detection modality for radio- frequency sensing with a double-resonance alignment magnetometer, Phys. Rev. Appl.22, 054033 (2024)

    2024

  84. [92]

    J. M. Robinson, M. Miklos, Y. M. Tso, C. J. Kennedy, T. Bothwell, D. Kedar, J. K. Thompson, and J. Ye, Direct comparison of two spin-squeezed optical clock ensembles at the 10 −17 level, Nature Physics20, 208 (2024)

    2024

  85. [93]

    Ferioli, A

    G. Ferioli, A. Glicenstein, I. Ferrier-Barbut, and A. Browaeys, A non-equilibrium superradiant phase tran- sition in free space, Nature Physics19, 1345 (2023)

    2023

  86. [94]

    Agarwal, E

    S. Agarwal, E. Chaparro, D. Barberena, A. P. n. Orioli, G. Ferioli, S. Pancaldi, I. Ferrier-Barbut, A. Browaeys, and A. Rey, Directional superradiance in a driven ultra- cold atomic gas in free space, PRX Quantum5, 040335 (2024)

    2024

  87. [95]

    Goncalves, L

    D. Goncalves, L. Bombieri, G. Ferioli, S. Pancaldi, I. Ferrier-Barbut, A. Browaeys, E. Shahmoon, and D. Chang, Driven-dissipative phase separation in free- space atomic ensembles, PRX Quantum6, 020303 (2025)

    2025

  88. [96]

    G´ orecki, F

    W. G´ orecki, F. Albarelli, S. Felicetti, R. Di Candia, and L. Maccone, Interplay between time and energy in bosonic noisy quantum metrology, PRX Quantum6, 020351 (2025)

    2025

  89. [97]

    Gammelmark and K

    S. Gammelmark and K. Mølmer, Fisher information and the quantum Cram´ er-Rao sensitivity limit of continuous measurements, Phys. Rev. Lett.112, 170401 (2014)

    2014

  90. [98]

    X. J. Liu, M. Ding, Y. Li, W. Hu, Y. H.and Jin, and J. C. Fang, Transverse relaxation determination based on light polarization modulation for spin-exchange relax- ation free atomic magnetometer, Chinese Physics B27, 073201 (2018)

    2018

  91. [99]

    S. P. Alvarez, P. Gomez, S. Coop, R. Zamora-Zamora, C. Mazzinghi, and M. W. Mitchell, Single-domain bose condensate magnetometer achieves energy resolution per bandwidth below ℏ, Proceedings of the National Academy of Sciences119, e2115339119 (2022), https://www.pnas.or...

    2022 doi

  92. [100]

    Engelhardt, J

    G. Engelhardt, J. Luo, V. M. Bastidas, and G. Platero, Photon-resolved Floquet theory. II. Open quantum sys- tems, Phys. Rev. A110, 063708 (2024)

    2024

  93. [101]

    Radaelli, G

    M. Radaelli, G. T. Landi, K. Modi, and F. C. Binder, Fisher information of correlated stochastic processes, New Journal of Physics25, 053037 (2023)

    2023

  94. [102]

    J. Kong, R. Jim´ enez-Mart ´ ınez, C. Troullinou, V. G. 29 Lucivero, G. T´ oth, and M. W. Mitchell, Measurement- induced, spatially-extended entanglement in a hot, strongly-interacting atomic system, Nature Communica- tions11, 2415 (2020)

    2020

  95. [103]

    Windt, M

    B. Windt, M. Bello, D. Malz, and J. I. Cirac, Effects of retardation on many-body superradiance in chiral waveg- uide qed, Phys. Rev. Lett.134, 173601 (2025)

    2025

  96. [104]

    Numerical data and source code can be downloaded from www.georg-engelhardt-research.de

  97. [105]

    A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Introduction to quantum noise, measurement, and amplification, Rev. Mod. Phys.82, 1155 (2010)

    2010

  98. [106]

    D. A. Bagrets and Y. V. Nazarov, Full counting statistics of charge transfer in Coulomb blockade systems, Phys. Rev. B67, 085316 (2003)

    2003

  99. [107]

    M. Li, L. JY., G. Platero, and G. Engelhardt, Full- counting statistics and quantum information of dispersive readout with a squeezed environment, to be publisched (2025)

    2025

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