Enhancing parameter estimation precision in dissipative environment with two-photon driving

Chunling Xu; Dong Xie

REVIEW 2 major objections 6 minor 54 references

Enhancing parameter estimation precision in dissipative environment with two-photon driving

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read Frequency uncertainty in a lossy cavity drops to zero at long encoding times when a two-photon drive is tuned to a damping-dependent threshold.

desk verdict Two-photon driving does protect long-time frequency estimation in a lossy cavity, and the final formula holds up, but the paper's printed intermediates are wrong by factors of two and the 'zero uncertainty' claim is an order-of-limits artifact. read the letter →

arxiv 1908.03838 v1 pith:VNEP3DEP submitted 2019-08-11 quant-ph

Dong Xie , Chunling Xu This is my paper · ORCID

classification quant-ph

keywords two-photondrivingfrequencyestimationquantummetrologydissipativeenvironmentPTsymmetryexceptionalpointhomodynedetectionerrorpropagation

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

In a lossy optical cavity, frequency information is normally destroyed: as the encoding time grows, the uncertainty of the estimated frequency diverges. This paper shows that adding a two-photon (parametric) drive changes the long-time behavior: the uncertainty saturates at a finite value, and when the drive strength λ satisfies λ² = γ² + ω² (γ the damping rate, ω the unknown frequency) the uncertainty approaches zero. The result holds under the Wigner–Weisskopf/Markov approximation for the environment and is obtained for two experimentally practical measurements, direct photon counting and homodyne detection. A sympathetic reader would take the paper to establish that two-photon driving can act as a noise-resisting resource for ultrasensitive frequency estimation under long encoding times.

What carries the argument

The central object is the two-photon driving term $(i\lambda/2)(\hat a^{\dagger2}-\hat a^2)$ in the Hamiltonian. In the Heisenberg-picture basis $(\hat a,\hat a^\dagger)^T$ it produces an effective non-Hermitian PT-symmetric Hamiltonian $\hat H_{\rm eff} = \begin{pmatrix} \omega & i\lambda \\ i\lambda & -\omega \end{pmatrix}$ with eigenvalues $\pm\sqrt{\omega^2-\lambda^2}$; the exceptional point $\lambda = \omega$ is where the two eigenvalues coalesce. Dissipation $\gamma$ shifts the long-time threshold to $\lambda^2 = \gamma^2 + \omega^2$, where the steady-state photon number diverges and the variance formula (via the fourth-order decoupling relation, Eq. 36) gives a vanishing estimation error. The two measurement schemes, direct photon detection $\hat a^\dagger\hat a$ and homodyne detection $M_h = (e^{-i\theta}\hat a^\dagger + e^{i\theta}\hat a)/2$, supply the observable whose mean and variance enter the error-propagation formula $\delta\omega = \delta M / |\partial M/\partial \omega|$.

What would settle it

Measure the photon-number distribution (or $\langle \hat a^{\dagger 2}\hat a^2\rangle$) in the steady state of a driven lossy cavity as $\lambda^2$ approaches $\gamma^2 + \omega^2$; if the Gaussian decoupling identity fails, the error-propagation formula changes. A direct experiment could also measure the estimated-frequency uncertainty as a function of $\lambda$ at long fixed time and compare its minimum and divergence to Eq. (13).

Watch

Extended reading notes

Core claim

The paper's central claim is that the error-propagation formula for photon-number detection gives δω² ≈ ((−λ² + γ² + ω²)²[−λ² + 3(γ² + ω²)])/(4λ²ω²) in the long-time limit (Eq. 13), while the same quantity without the drive (λ = 0) is infinite. Consequently, for ω ≠ 0, δω → 0 as λ² → γ² + ω²: the drive strength that balances damping and frequency acts as a threshold at which measurement precision becomes ultra-sensitive, in stark contrast with the undriven case. The paper also treats the λ² > γ² + ω² regime with a coherent initial state, showing δω² scales as 1/(N t²), and identifies the specific magnitude λ = √(γ² + ω²) as optimal when γ ≪ ω. The mechanism is the effective non-Hermitian PT-symmetric dynamics generated by the drive; the exceptional point at λ ≈ ω is the noiseless limit of the threshold.

Load-bearing premise

The result hinges on the fourth-order decoupling relation, which treats the photon-number fluctuations as nearly Gaussian; if the actual steady state near threshold is non-Gaussian, the predicted cancellation of the uncertainty is not guaranteed.

Editorial extensions

If this is right

Without the drive ($\lambda = 0$) the long-encoding-time uncertainty diverges; with any nonzero drive it becomes finite, so dissipation no longer destroys frequency information completely.
When $\lambda^2$ approaches $\gamma^2 + \omega^2$, the uncertainty approaches zero, giving a concrete operating point for ultrasensitive measurement; the precision is independent of the initial field state for direct photon detection in the small-drive regime.
For $\lambda^2 > \gamma^2 + \omega^2$ with a bright coherent seed, direct detection gives $\delta\omega^2 \sim 1/(N t^2)$, restoring the ideal-metrology scaling, and homodyne detection at phase $\theta = \pi/2$ resolves small frequencies.
The near-zero uncertainty is bought at the cost of a diverging steady-state photon count, so for fixed energy the optimal drive will sit slightly below the threshold.
Because the argument treats only photon-counting and homodyne observables, it establishes an experimentally friendly precision enhancement without needing an optimal quantum measurement.

Reading between the lines

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

Editorial inference: a resource-normalized version of the result would put the practical optimum below the threshold, because the photon number diverges at $\lambda^2 = \gamma^2 + \omega^2$; weighting $\delta\omega^2$ by the average photon number would replace the zero with a finite minimum.
Editorial inference: the same drive should protect estimation of other parameters such as the damping rate $\gamma$ or a detuning, and the framework naturally extends to parametrically driven Bose systems beyond optics, which the authors note as future work.
Editorial inference: the decoupling approximation can be checked numerically by solving the Lindblad master equation near threshold; a non-Gaussian steady state would alter the variance and likely cap the precision improvement.

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.

Desk Editor's Note

read the letter

Quick take: the paper's central effect is real. A two-photon drive (parametric amplification) below threshold keeps the mean photon number finite in the long-time limit, so a direct photon count carries information about the frequency omega that would otherwise vanish. The closed-form uncertainty formula in Eq. (13) matches an independent re-derivation from the Appendix integrals. That is the actual new content: simple error-propagation expressions for frequency estimation in a damped, driven cavity, with a PT-symmetry interpretation.

What works: the model is the standard Wigner–Weisskopf/Langevin treatment, and the approximations leading to Eq. (10) are standard. The Gaussian decoupling relation in Eq. (36) is not the weak spot—because the Hamiltonian is quadratic and the bath is in vacuum, the four-operator expectation is exact Wick, not an approximation. The authors also deserve credit for working out the below-threshold, above-threshold, and at-threshold cases separately and for comparing direct photon detection with homodyne detection.

Where it gets shaky. First, the displayed derivation has internal factor-of-two errors. Eq. (11) gives M_d = lambda^2/D, but the integrals in Eqs. (37)–(38) give lambda^2/(2D). Eq. (12) is off by a factor of four. These errors cancel in the final Eq. (13), which is why the final answer is right, but as printed the derivation from Eq. (10) through Eq. (13) is not reproducible.

Second, and more important, the 'delta_omega → 0 near threshold' headline is a limit that does not commute. The long-time expression Eq. (10) is only valid for t[gamma − sqrt(lambda^2 − omega^2)] ≫ 1, which near threshold requires D = gamma^2 + omega^2 − lambda^2 ≫ gamma/t. If you take D → 0 at fixed t, you leave the asymptotic regime. The actual finite-time optimum sits at D ~ 1/t and gives delta_omega ~ 1/t, which is the ideal-metrology scaling—a good result, but not zero. The paper should say this explicitly instead of letting the reader think the uncertainty vanishes at any large but finite time.

Third, the manuscript is littered with typos (r for gamma, w for omega, missing brackets, mislabeled equations). That makes an already technical read harder than it needs to be. The citation pattern itself is reasonable; the key noisy-metrology and exceptional-point sensor papers are cited.

Verdict: the physics is sound, the final formula is correct, and the effect is worth knowing. But the paper needs careful revision before I would rely on its derivation. It deserves a serious referee, not a desk reject, and the referee should ask for corrected intermediate equations and an explicit discussion of the order of limits.

Referee Report

2 major / 6 minor

Summary. The manuscript studies frequency estimation of a single optical cavity mode subject to Markovian dissipation and a two-photon parametric drive. The authors solve the Heisenberg-Langevin equations under the Wigner-Weisskopf/Markov approximation, derive a long-time expression for the field operator, and evaluate the error-propagation uncertainty for direct photon-number detection and homodyne detection. The main claimed result is that in the long-encoding-time limit the estimation uncertainty of the frequency becomes finite and can approach zero as the two-photon drive amplitude approaches λ²=γ²+ω², in contrast to the no-drive case where the uncertainty diverges. The paper also gives results for large and exactly critical drive amplitudes and interprets the effect through an effective PT-symmetric Hamiltonian.

Significance. If the central result is correct, it is a useful and non-obvious observation: a continuous two-photon drive can counteract Markovian loss in frequency estimation and restore a sensitivity that improves with time, apparently without requiring engineered spectral densities or entangled probes. The calculations are analytical and use no fitted parameters, and the proposed measurement schemes are experimentally accessible. The PT-symmetry interpretation is suggestive. However, the significance is conditional on correcting internal inconsistencies in the long-time derivation and on stating precisely the order of limits behind the "zero uncertainty" claim.

major comments (2)

[Section II.A, Eqs. (11)-(12) and Appendix B, Eqs. (37)-(39)] The displayed derivation from Eq. (10) to Eq. (13) is not reproducible as written. Equation (11) states M_d=λ²/(γ²+ω²−λ²), but the retained terms in Eq. (10), combined with the integrals in Eqs. (37)-(38), give ⟨a†a⟩=λ²/[2(γ²+ω²−λ²)]. Likewise, Eq. (12) is not obtained from Eqs. (37)-(39) with the field operator in Eq. (10). The final result in Eq. (13) may be qualitatively correct, but the intermediate expressions must be corrected and the derivation rewritten with consistent factors.
[Section II.A, after Eq. (10) and Eq. (13)] The central zero-uncertainty claim relies on a nonuniform order of limits. The asymptotic condition stated before Eq. (10), t[γ−(λ²−ω²)^{1/2}]≫1, becomes near threshold t(γ²+ω²−λ²)/(2γ)≫1. Equation (13) is obtained by first taking t→∞ at fixed λ and then sending λ²→γ²+ω². For any fixed finite encoding time t, the detuning D=γ²+ω²−λ² must remain ≳1/t for Eq. (10) to be valid, so δω cannot be made arbitrarily small; the best achievable scaling is δω∼1/t. The abstract and conclusion should state the order of limits explicitly and discuss the finite-time trade-off between precision and the validity of the asymptotic expansion.

minor comments (6)

[Appendix A] Appendix A states K(t−s)=πJ(ω)δ(t−s), which is inconsistent with Eq. (6) and the definition γ=πJ(ω); with the quoted convention the Wigner-Weisskopf reduction gives K=2γδ(t−s).
[Equations (13), (15), (40)-(43)] The typesetting uses "w" or "w2" where ω or ω² is intended; these should be corrected throughout.
[Appendix B, Eq. (36)] The decoupling relation in Eq. (36) is presented as an approximation, but for the field operator in Eq. (10), which is linear in the bath operators with the bath initially in vacuum, the Wick contraction is exact; stating this would remove the impression of an uncontrolled Gaussian assumption.
[Section III] The PT-symmetry discussion refers to the exceptional point ω=λ of the effective Hamiltonian in Eq. (24), whereas the relevant threshold in the dissipative analysis is λ=√(γ²+ω²); the connection is heuristic and should be labeled as such.
[Section II.B and Appendix B, Eq. (41)] The simplification from the full expression in Eq. (41) to the displayed result in Eq. (15) is not shown; the reader cannot verify the reduction, and the claim that the uncertainty increases with λ should be justified from the full expression.
[References] The reference list contains formatting errors, for example reference [21] is merged with the following citation, and some entries are incomplete; the bibliography should be checked carefully.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the precision formulas are computed from the stated Hamiltonian and stated approximations, with no fitted input renamed as a prediction.

full rationale

The paper's central result, Eq. (13), is obtained by solving the Heisenberg equations for the stated Hamiltonian (1) under the Wigner-Weisskopf/Markov approximation, substituting the asymptotic field operator (10) into the error-propagation formula (8) with the photon-number measurement operator, and then evaluating the moments in Appendix B. No parameter is fitted to any subset of data and no target quantity is inserted by hand: the uncertainty δω is explicitly computed from ⟨a†a⟩ and its variance, which are functions of the model parameters λ, γ, and ω. The decoupling relation (36) is a stated statistical approximation for four-operator expectations, not a definition of the predicted uncertainty or a restatement of Eq. (13). The PT-symmetry discussion in Section III is interpretive and is not used to derive the quantitative results. The only self-citation is Ref. [33], which appears as background on correlated environments and is not load-bearing for the derivation. Any concern about the non-uniform t→∞ and λ→√(γ²+ω²) limits, or the apparent factor-of-2 mismatch between Eq. (11) and the integrals (37)-(38), is a correctness or reproducibility issue, not circularity: those equations do not assume the conclusion they are used to derive. The paper therefore does not reduce its claim to its inputs by construction.

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

The derivation rests on standard Markovian open-quantum-system tools and the Gaussian decoupling approximation; no free parameters are fitted and no new entities are introduced.

assumptions (6)

domain assumption Wigner-Weisskopf (Markov) approximation: the spectral density J(omega') is replaced by its value at the mode frequency and the frequency integral is extended to minus infinity, giving K(t-s) = 2 gamma delta(t-s).
Used in Section II to convert the integro-differential equations (2)-(5) into Markovian linear equations and obtain closed-form G(t) and L(t).
domain assumption Environment is initially in the vacuum state |Psi_E(0)> = |{0_k}>.
Used to compute expectation values over the environment; non-vacuum thermal noise would change the photon statistics and hence the error formula.
domain assumption Fourth-order decoupling (Gaussian moment) relation, Eq. (36), is valid for the computation of the photon-number variance.
Used in Appendix B to evaluate delta_M_d squared; exact only for Gaussian states, which the linear Heisenberg dynamics supports in the long-time limit.
domain assumption The error propagation formula delta_omega = delta_M over the absolute value of the derivative of M with respect to omega is used instead of the quantum Fisher information; the chosen direct and homodyne detections are not assumed optimal.
The paper explicitly states the detection schemes are not optimal, so the results are sensitivity bounds for these schemes, not fundamental limits.
domain assumption The model Hamiltonian (1) adequately describes the two-photon driving of a single cavity mode in a dissipative environment.
Starting point of the paper; the two-photon drive term is treated as an ideal degenerate parametric amplifier.
standard math Bose commutation relation [a(t), a dagger(t)] = 1 is preserved, leading to the normalization |G|^2 + |L|^2 + sum over k of (|mu_k|^2 + |nu_k|^2) = 1.
Used in Appendix A to ensure the Heisenberg-picture expansion is canonical.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Enhancing parameter estimation precision in dissipative environment with two-photon driving." pith.science (2026). https://pith.science/paper/VNEP3DEP

@misc{pith2026190803838,
  author       = {Pith},
  title        = {Pith review of: Enhancing parameter estimation precision in dissipative environment with two-photon driving},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VNEP3DEP}},
  note         = {Machine review of arXiv:1908.03838}
}

read the original abstract

We investigate the frequency estimation of an optical field suffering from an unavoidable dissipative environment. Generally, dissipative noises greatly reduce the precision. Here, we find that two-photon driving can improve the measurement precision by resisting the noises. Moreover, in long time, the uncertainty of frequency can be close to 0 with a proper magnitude of the parametric two-photon drive, which is in sharp contrast to the uncertainty going to infinity without the two-photon driving. Our results show that two-photon driving can realize the ultrasensitive measurement in dissipative environment under the long-encoding-time condition.

Figures

Figures reproduced from arXiv: 1908.03838 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

54 extracted references · 52 canonical work pages

[1]

So considering the cost of energy (or time), the optimal estimation precision of ω should not be equal to 0 exactly

The reason is that when the magnitude is close to √ ω 2 +γ 2, the number of exciting photons ⟨ˆa†(t)ˆa(t)⟩ is close to inﬁnity. So considering the cost of energy (or time), the optimal estimation precision of ω should not be equal to 0 exactly. IV. CONCLUSION AND OUTLOOK We have investigated the function of the two-photon drive on impro ving the estimatio...

work page
[2]

306, 1330 (2004)

Giovanetti V., Lloyd S., Maccone L.: Quantum-enhanced m easurements: beating the standard quantum limit Science. 306, 1330 (2004)

work page 2004
[3]

Giovannetti V., Lloyd, S., and Maccone L.: Quantum Metro logy. Phys. Rev. Lett. 96, 010401 (2006)

work page 2006
[4]

A.: High-order inert ial phase shifts for time-domain atom interferometers

Bongs, K., Launay, R., Kasevich, M. A.: High-order inert ial phase shifts for time-domain atom interferometers. App l. Phys. B 84, 599 (2006)

work page 2006
[5]

Paris, M. G. A.: Quantum Estimation for Quantum Technolo gy. Int. J. Quant. Inf. 7, 125 (2009)

work page 2009
[6]

A., Janousek, J., Daria, V., Knittel, J., Hage , B., Bachor, H.-A., Bowen, W

Taylor, M. A., Janousek, J., Daria, V., Knittel, J., Hage , B., Bachor, H.-A., Bowen, W. P.: Biological measurement be yond the quantum limit. Nat. Photon. 7, 229 (2013)

work page 2013
[7]

Slussarenko, M

S. Slussarenko, M. M. Weston, H. M. Chrzanowski, L. K. Sha lm, V. B. Verma, S. W. Nam, and G. J. Pryde, Nat. Photonics 11, 700 (2017)

work page 2017
[8]

M. Xiao, L. A. Wu, and H. J. Kimble, Precision Measurement Beyond the Shot-Noise Limit, Phys. Rev. Lett. 59, 278 (1987)

work page 1987

Show all 54 references

[9]

Grangier, R

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, Sque ezed-Light-Enhanced Polarization Interferometer, Phys. Rev. Lett. 59, 2153 (1987)

1987
[10]

Lange, J

K. Lange, J. Peise, B. L¨ ucke, L. Pezz`e, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Kl empt, Improvement of an Atomic Clock Using Squeezed Vacuum, Phys. Rev. Lett. 11 7, 143004 (2016)

2016
[11]

Schnabel, N

R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. La m, Quantum metrology for gravitational wave astronomy, Nat . Commun. 1, 121 (2010)

2010
[12]

Aasi et al., Enhanced sensitivity of the LIGO gravita tional wave detector by using squeezed states of light, Nat

J. Aasi et al., Enhanced sensitivity of the LIGO gravita tional wave detector by using squeezed states of light, Nat. Photonics 7, 613 (2013)

2013
[13]

Cappellaro, J

P. Cappellaro, J. Emerson, N. Boulant, C. Ramanathan, S . Lloyd, and D. G. Cory, Phys. Rev. Lett. 94, 020502 (2005)

2005
[14]

Nagata, R

T. Nagata, R. Okamoto, J. L. OBrien, K. Sasaki, and S. Tak euchi, Science 316, 726 (2007)

2007
[15]

Israel, S

Y. Israel, S. Rosen, and Y. Silberberg, Phys. Rev. Lett. 112, 103604 (2014)

2014
[16]

Luo, Y.-Q

X.-Y. Luo, Y.-Q. Zou, L.-N. Wu, Q. Liu, M.-F. Han, M. K. Te y, and L. You, Science 355, 620 (2017)

2017
[17]

Mankei Tsang, Conservative classical and quantum reso lution limits for incoherent imaging, Journal of Modern Opt ics, 65:11, 1385-1391 (2018)

2018
[18]

Barzanjeh, S

S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Sha piro, and S. Pirandola, Phys. Rev. Lett. 114, 080503 (2015)

2015
[19]

C. W. Sandbo Chang, A. M. Vadiraj, J. Bourassa, B. Balaji , and C. M. Wilson, Appl. Phys. Lett. 114 112601 (2019)

2019
[20]

Grote, K

H. Grote, K. Danzmann, K. L. Dooley, R. Schnabel, J. Slut sky, and H. Vahlbruch, First Long-Term Application of Squee zed States of Light in a Gravitational-Wave Observatory, Phys. Rev. Lett. 110, 181101
[21]

Lisa Barsotti, Jan Harms, and Roman Schnabel, Squeezed vacuum states of light for gravitational wave detectors,
[22]

Rep. Prog. Phys. 82 016905 (2019). U. Dorner, R. Demkowi cz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, Phys. Rev. Lett. 102, 040403 (2 009)

2019
[23]

Demkowicz-Dobrzanski, U

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. L undeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, Phys . Rev. A 80, 013825 (2009)

2009
[24]

Hudelist, J

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhan g, Nature Communications 5, 3049 (2014)

2014
[25]

Y.-S. Wang, C. Chen, and J.-H. An, New Journal of Physics 19, 113019 (2017)

2017
[26]

Margret Heinze and Robert K¨ onig, Phys. Rev. Lett. 123, 010501 (2019)

2019
[27]

Qi Yao, Jun Zhang, Xiao-Feng Yi, Li You, and Wenxian Zhan g, Phys. Rev. Lett. 122, 010408 (2019)

2019
[28]

Lidar, Phys

Lian-Ao Wu and Daniel A. Lidar, Phys. Rev. A 70, 062310 (2 004)
[29]

Lidar, Phys

Asoka Biswas and Daniel A. Lidar, Phys. Rev. A 74, 062303 (2006)

2006
[30]

Lidar, Phys

Bibek Pokharel, Namit Anand, Benjamin Fortman, and Dan iel A. Lidar, Phys. Rev. Lett. 121, 220502 (2018)

2018
[31]

Vitali and P

D. Vitali and P. Tombesi, Phys. Rev. A 59, 4178 (1999)

1999
[32]

Vitali and P

D. Vitali and P. Tombesi, Phys. Rev. A 65, 012305 (2001)

2001
[33]

A. W. Chin, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett . 109, 233601 (2012)

2012
[34]

Dong Xie, An Min Wang, Phys. Lett. A 378, 2079-2084 (2014 )

2014
[35]

J. J. Cooper, D. W. Hallwood, J. A. Dunningham, and J. Bra nd, Phys. Rev. Lett. 108, 130402 (2012)

2012
[36]

X.-M. Lu, S. Yu, and C. H. Oh, Nature Communications 6, 72 82 (2015)

2015
[37]

Yu-Xin Wang and A. A. Clerk, Phys. Rev. A 99, 063834 (2019 ). J. Wiersig, Phys. Rev. Lett. 112, 203901 (2014)

2014
[38]

Zhang et al., Sci

N. Zhang et al., Sci. Rep. 5, 11912 (2015)

2015
[39]

Wiersig, Phys

J. Wiersig, Phys. Rev. A 93, 033809 (2016)

2016
[40]

Ren et al., Opt

J. Ren et al., Opt. Lett. 42, 1556 (2017). 9

2017
[41]

Chen et al., Nature 548, 192 (2017)

W. Chen et al., Nature 548, 192 (2017)

2017
[42]

Hodaei et al., Nature 548, 187 (2017)

H. Hodaei et al., Nature 548, 187 (2017)

2017
[43]

Ono and H

T. Ono and H. F. Hofmann, Phys. Rev. A 81, 033819 (2010)

2010
[44]

P. A. Knott, T. J. Proctor, K. Nemoto, J. A. Dunning-ham, and W. J. Munro, Phys. Rev. A 90, 033846 (2014)

2014
[45]

Dorner, R

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. L undeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, Phys . Rev. Lett. 102, 040403 (2009)

2009
[46]

A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fish er, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987)

1987
[47]

Weisskopf and E

V. Weisskopf and E. Wigner, Eur. Phys. J. A 63, 54 (1930)

1930
[48]

Louisell, Quantum Statistical Properties of Radiat ion (Wiley, New York, 1973), pp

W. Louisell, Quantum Statistical Properties of Radiat ion (Wiley, New York, 1973), pp. 418-428

1973
[49]

L.-P. Yang, C. Y. Cai, D. Z. Xu, W.-M. Zhang, and C. P. Sun, Phys. Rev. A 87, 012110 (2013)

2013
[50]

Wiseman and G

H. Wiseman and G. Milburn, quantum measurement and cont rol Cambridge University Press, New York, 2010)

2010
[51]

Huelga, C

S.F. Huelga, C. Macchiavello, T. Pellizzari, A.K. Eker t, M.B. Plenio, J.I. Cirac, Phys. Rev. Lett. 79 (1997) 3865

1997
[52]

Longhi, Opt

S. Longhi, Opt. Lett. 43, 5371 (2018)

2018
[53]

Langbein, Phys

W. Langbein, Phys. Rev. A 98, 023805 (2018)

2018
[54]

Naikoo, K

J. Naikoo, K. Thapliyal, A. Pathak, and S. Banerjee, Phy s. Rev. A 97, 063840 (2018)

2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

So considering the cost of energy (or time), the optimal estimation precision of ω should not be equal to 0 exactly

The reason is that when the magnitude is close to √ ω 2 +γ 2, the number of exciting photons ⟨ˆa†(t)ˆa(t)⟩ is close to inﬁnity. So considering the cost of energy (or time), the optimal estimation precision of ω should not be equal to 0 exactly. IV. CONCLUSION AND OUTLOOK We have investigated the function of the two-photon drive on impro ving the estimatio...

work page

[2] [2]

306, 1330 (2004)

Giovanetti V., Lloyd S., Maccone L.: Quantum-enhanced m easurements: beating the standard quantum limit Science. 306, 1330 (2004)

work page 2004

[3] [3]

Giovannetti V., Lloyd, S., and Maccone L.: Quantum Metro logy. Phys. Rev. Lett. 96, 010401 (2006)

work page 2006

[4] [4]

A.: High-order inert ial phase shifts for time-domain atom interferometers

Bongs, K., Launay, R., Kasevich, M. A.: High-order inert ial phase shifts for time-domain atom interferometers. App l. Phys. B 84, 599 (2006)

work page 2006

[5] [5]

Paris, M. G. A.: Quantum Estimation for Quantum Technolo gy. Int. J. Quant. Inf. 7, 125 (2009)

work page 2009

[6] [6]

A., Janousek, J., Daria, V., Knittel, J., Hage , B., Bachor, H.-A., Bowen, W

Taylor, M. A., Janousek, J., Daria, V., Knittel, J., Hage , B., Bachor, H.-A., Bowen, W. P.: Biological measurement be yond the quantum limit. Nat. Photon. 7, 229 (2013)

work page 2013

[7] [7]

Slussarenko, M

S. Slussarenko, M. M. Weston, H. M. Chrzanowski, L. K. Sha lm, V. B. Verma, S. W. Nam, and G. J. Pryde, Nat. Photonics 11, 700 (2017)

work page 2017

[8] [8]

M. Xiao, L. A. Wu, and H. J. Kimble, Precision Measurement Beyond the Shot-Noise Limit, Phys. Rev. Lett. 59, 278 (1987)

work page 1987