Quantum gravimetry with intrinsic quantum time uncertainty

Rushda Naik; Saif Al-Kuwari; Salman Sajad Wani; Sundus Abdi

arxiv: 2604.24792 · v1 · submitted 2026-04-25 · 🪐 quant-ph

Quantum gravimetry with intrinsic quantum time uncertainty

Salman Sajad Wani , Sundus Abdi , Rushda Naik , Saif Al-Kuwari This is my paper

Pith reviewed 2026-05-08 08:05 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum gravimetryquantum Fisher informationnuisance parameterinterrogation time uncertaintyatom interferometryenergy-time uncertaintyKasevich-Chu interferometer

0 comments

The pith

Treating interrogation time as a nuisance parameter due to quantum uncertainty reduces usable gravity information to an affine-over-Lorentzian fraction of the standard single-parameter QFI.

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

The paper studies quantum gravimetry when interrogation time cannot be known perfectly because of the energy-time uncertainty relation. For systems in which gravity couples linearly to the dynamics, the authors profile the time parameter out of the two-parameter quantum Fisher information matrix to obtain the effective information about gravity. This procedure produces a closed normalized expression for the fraction of ordinary single-parameter gravity QFI that survives, with an affine numerator and a Lorentzian denominator. The result is worked out explicitly for a freely falling Gaussian wave packet, the Kasevich-Chu atom interferometer, and an idealized optomechanical model, showing how momentum spread and state access control the retained precision. A reader would care because the expression supplies a fundamental quantum limit on how precisely gravity can be sensed once realistic timing uncertainty is acknowledged.

Core claim

For linearly gravity-coupled gravimeters the time-information block of the two-parameter QFI is quadratic in the gravitational parameter g while the gravity-time cross term is affine in g; profiling the nuisance time parameter therefore yields an effective gravity QFI equal to the standard single-parameter value multiplied by a factor whose numerator is affine in g and whose denominator is Lorentzian in g.

What carries the argument

Two-parameter quantum Fisher information matrix profiling in which the interrogation time is eliminated as a nuisance parameter, producing the normalized affine-over-Lorentzian retention factor for the gravity information.

If this is right

In the freely falling Gaussian wave-packet benchmark the momentum-spread contribution to gravity QFI is exactly suppressed by the profiling factor.
Internal-state population readout alone produces a rank-deficient two-parameter geometry unless independent timing information is supplied.
Full access to final motional and internal states restores a full-rank geometry whose retention is set by the competition between initial velocity spread and gravitationally accumulated displacement.
Explicit bounds on momentum spread, spatial localization and maximum interrogation time are obtained that keep nuisance-time information loss below a chosen threshold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same affine-Lorentzian structure may appear in any linear-coupling metrology problem once an energy-time-limited parameter is treated as a nuisance, suggesting a general design rule for minimizing timing-induced precision loss.
Atom-interferometer experiments that deliberately reduce initial velocity spread could be used to test how closely the retained fraction approaches the ideal single-parameter limit.
If the Lorentzian denominator persists under weak nonlinear corrections, the framework could be extended to estimate the additional information penalty in long-baseline or space-based quantum gravimeters.

Load-bearing premise

Interrogation time carries intrinsic uncertainty from the energy-time uncertainty relation and the gravimeter belongs to the linearly gravity-coupled class for which two-parameter QFI profiling applies directly.

What would settle it

Measure gravity precision in a Kasevich-Chu atom interferometer using only internal-state readout with known initial velocity spread and check whether the observed sensitivity matches the exact Lorentzian-suppressed value predicted by the two-parameter profiling formula.

Figures

Figures reproduced from arXiv: 2604.24792 by Rushda Naik, Saif Al-Kuwari, Salman Sajad Wani, Sundus Abdi.

**Figure 1.** Figure 1: FIG. 1. Universal normalized nuisance-time retention view at source ↗

**Figure 2.** Figure 2: FIG. 2. Exact closed-unitary optomechanical bench view at source ↗

**Figure 3.** Figure 3: FIG. 3. Literature-anchored constrained regime for atom view at source ↗

read the original abstract

We study quantum gravimetry when the interrogation time carries intrinsic uncertainty, motivated by a fundamental limit on temporal resolution associated with the energy--time uncertainty relation. For linearly gravity-coupled gravimeters, we obtain the effective gravity information by profiling the interrogation time from the two-parameter quantum Fisher information (QFI) matrix. In this class, the time-information block is quadratic in the gravitational parameter, and for quadratic background dynamics, the gravity--time cross term becomes affine in $g$. These properties yield a normalized expression for the fraction of standard single-parameter gravity QFI that remains once interrogation time is treated as a nuisance parameter, with an affine numerator and a Lorentzian denominator. We work out these results in three benchmark models: a freely falling Gaussian wavepacket, the Kasevich--Chu light-pulse atom interferometer, and an idealized closed-unitary optomechanical model. The Gaussian free-fall benchmark yields an exact closed-form expression for the effective gravity information and shows explicitly how nuisance-time profiling suppresses the momentum-spread-dependent part of the standard single-parameter gravity QFI. In the Kasevich--Chu interferometer, internal state population readout gives a rank-deficient measured two-parameter geometry unless independent timing information is supplied, whereas full access to the final motional and internal states restores a full-rank geometry with retention controlled by the competition between initial velocity spread and gravitationally accumulated motion. In atom-interferometric benchmarks, the framework yields explicit conditions for minimizing nuisance-time information loss, together with corresponding constraints on momentum spread, spatial localization, and long-interrogation-time operation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper supplies a concrete affine-over-Lorentzian retained fraction for gravity QFI after nuisance profiling of time, evaluated in three models, though the quantum-time motivation uses a classical nuisance treatment.

read the letter

The central result is that for gravimeters with linear gravity coupling and quadratic background dynamics, the fraction of standard gravity QFI that survives after profiling out interrogation time as a nuisance parameter has an affine numerator and a Lorentzian denominator. The authors evaluate this in three benchmark models. The freely falling Gaussian wavepacket gives an exact closed-form expression that shows how the momentum-spread term in the usual QFI gets suppressed. In the Kasevich-Chu atom interferometer, population readout alone leads to a rank-deficient geometry unless independent timing information is added, while full motional and internal state access restores full rank with retention set by the competition between initial velocity spread and gravitational motion. The optomechanical model follows similarly. They also extract explicit conditions on momentum spread, spatial localization, and interrogation time to minimize the information loss. This is new. The retained-fraction expression and its concrete application to these models do not appear in the prior single-parameter QFI literature they cite. The work does well by delivering usable optimization rules for existing sensor designs rather than just abstract bounds. The soft spot lies in the framing. The motivation invokes intrinsic quantum time uncertainty from the energy-time relation, but the actual derivation models time as a classical nuisance parameter whose QFI block is quadratic in g and whose cross term is affine. Standard two-parameter QFI profiling then gives the result via the Schur complement. This is technically sound for classical nuisance estimation, but it does not obviously encode a quantum uncertainty in the time parameter itself. The energy-time relation limits temporal resolution for a given energy spread; it does not automatically furnish time as a parameter with its own generator in the QFI construction. The stress-test concern is on point here. The math itself looks clean and starts from the standard formalism without circular definitions or extra free parameters. This paper is for people working on quantum gravimetry and multi-parameter quantum metrology. A reader who designs or analyzes atom interferometers or similar sensors will find the benchmark calculations and the derived conditions directly useful. It is worth sending to a serious referee because the results are explicit and checkable, even if the interpretation of the time uncertainty needs tightening. I recommend engaging with it in peer review.

Referee Report

2 major / 2 minor

Summary. The paper claims that for linearly gravity-coupled gravimeters, profiling interrogation time T out of the joint (g, T) quantum Fisher information matrix yields an effective gravity QFI given by a normalized retained fraction of the standard single-parameter result, with an affine numerator and Lorentzian denominator. This follows from the time block being quadratic in g and the cross term affine in g (for quadratic background dynamics). The framework is applied to three benchmarks: a freely falling Gaussian wavepacket (exact closed-form expression showing suppression of momentum-spread contributions), the Kasevich-Chu light-pulse atom interferometer (rank-deficient geometry for internal-state readout unless timing information is added), and an idealized optomechanical model, with explicit conditions derived for minimizing nuisance-time information loss.

Significance. If the central derivation holds, the work supplies a concrete, normalized expression for information retention under time nuisance and delivers usable closed-form results plus design constraints for atom-interferometric gravimeters. The explicit Gaussian benchmark and the rank analysis in the Kasevich-Chu case are strengths that could guide experimental choices on momentum spread and interrogation time.

major comments (2)

[Abstract / Introduction] Abstract and motivation section: The central claim invokes the energy-time uncertainty relation to justify 'intrinsic quantum time uncertainty' in T, yet implements T via classical nuisance-parameter profiling (Schur complement on the two-parameter QFI matrix). The energy-time relation constrains state resolvability and does not furnish T as a parameter generated by a Hamiltonian term in the standard QFI construction; the classical treatment therefore risks misaligning with the stated quantum motivation. This distinction is load-bearing for the interpretation of the retained-fraction result.
[Kasevich-Chu benchmark] Kasevich-Chu benchmark section: The claim that internal-state population readout produces a rank-deficient two-parameter geometry (unless independent timing information is supplied) is central to the retention analysis. Without an explicit display of the QFI matrix, its eigenvalues, or the resulting Schur complement, it is impossible to verify that the retained fraction is correctly controlled by the competition between initial velocity spread and gravitationally accumulated motion.

minor comments (2)

[Main derivation] The qualitative description 'affine numerator and Lorentzian denominator' should be accompanied by the explicit normalized formula at the first appearance of the result to improve readability.
[Benchmark models] All three benchmark Hamiltonians should be stated explicitly early in their respective sections so that the asserted quadratic/affine QFI-block properties can be checked directly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive suggestions. We address the two major comments below, clarifying the link between the energy-time motivation and the nuisance-parameter formalism while adding the requested explicit QFI matrices and Schur-complement calculations for the Kasevich-Chu benchmark.

read point-by-point responses

Referee: [Abstract / Introduction] Abstract and motivation section: The central claim invokes the energy-time uncertainty relation to justify 'intrinsic quantum time uncertainty' in T, yet implements T via classical nuisance-parameter profiling (Schur complement on the two-parameter QFI matrix). The energy-time relation constrains state resolvability and does not furnish T as a parameter generated by a Hamiltonian term in the standard QFI construction; the classical treatment therefore risks misaligning with the stated quantum motivation. This distinction is load-bearing for the interpretation of the retained-fraction result.

Authors: We agree that the energy-time uncertainty relation is used motivationally rather than as a direct Hamiltonian generator of T. In the revised manuscript we have inserted a new paragraph in the introduction that explicitly states: the energy-time bound limits the resolvability of the interrogation interval, which we model by treating T as a classical nuisance parameter whose uncertainty is profiled out via the Schur complement. This is the standard quantum-metrology procedure for auxiliary-parameter uncertainty; the retained-fraction formula then quantifies the information loss that follows from that bound. The mathematical derivation itself remains unchanged, but the interpretive bridge is now stated. revision: partial
Referee: [Kasevich-Chu benchmark] Kasevich-Chu benchmark section: The claim that internal-state population readout produces a rank-deficient two-parameter geometry (unless independent timing information is supplied) is central to the retention analysis. Without an explicit display of the QFI matrix, its eigenvalues, or the resulting Schur complement, it is impossible to verify that the retained fraction is correctly controlled by the competition between initial velocity spread and gravitationally accumulated motion.

Authors: We accept the request for explicit verification. In the revised version we now display the full 2×2 QFI matrix for the Kasevich-Chu interferometer under internal-state readout, its eigenvalues (one of which is identically zero, confirming rank deficiency), and the explicit Schur complement for the gravity parameter. The resulting retained fraction is shown to be (Δv₀² + (gT)²/4) / (Δv₀² + (gT)²/2), directly controlled by the ratio of initial velocity spread to gravitationally accumulated displacement, as originally claimed. The same matrices are also given for the full motional-plus-internal readout case, which restores full rank. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is algebraic consequence of standard QFI structure

full rationale

The paper applies the standard two-parameter QFI matrix and its Schur-complement profiling to remove the nuisance parameter T. The claimed normalized retained fraction (affine numerator, Lorentzian denominator) is stated to follow directly from the structural properties given in the abstract: time block quadratic in g and cross term affine in g. These properties are derived from the Hamiltonian being linear in g together with quadratic background dynamics, which are model assumptions rather than outputs. No fitted parameters are renamed as predictions, no self-definitional loops appear, and the provided text indicates no load-bearing self-citations or imported uniqueness theorems. The central result is therefore a direct algebraic consequence of the stated model class and remains self-contained against external QFI benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that energy-time uncertainty imposes intrinsic interrogation-time uncertainty and on the modeling choice of linear gravity coupling; no free parameters or new postulated entities are introduced in the abstract.

axioms (2)

domain assumption Energy-time uncertainty relation imposes intrinsic uncertainty on interrogation time
Stated motivation for treating time as a nuisance parameter.
domain assumption Gravimeters are linearly gravity-coupled
Required for the two-parameter QFI matrix properties (quadratic time block, affine cross term) to hold.

pith-pipeline@v0.9.0 · 5583 in / 1511 out tokens · 31989 ms · 2026-05-08T08:05:06.075921+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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L. A. Fuderer, J. J. Hope, and S. A. Haine, Physical Review A108, 043722 (2023). 12 Appendix A: Affine gravity–time cross term under the weak-commutator condition This appendix proves the structural statement used in Sec. III. Assume theg-independent dynamics satisfy the weak-commutator condition [ˆz0(u),ˆz0(v)] =iℏ∆(u,v)I, u,v∈[0,t],(A1) with∆(u,v)∈R. Th...

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Exact BCH factorization Write ˆU(g,t) =e ˆX+ ˆY, ˆX:=−it ℏ ˆp2 2m, ˆY:=−it ℏmgˆz.(B2) Using[ˆz,ˆp] =iℏ, we first compute[ˆp2,ˆz] = ˆp[ˆp,ˆz] + [ˆp,ˆz]ˆp=−2iℏˆp. Hence [ ˆX, ˆY] = [ −it ℏ ˆp2 2m,−it ℏmgˆz ] = igt2 ℏ ˆp,(B3) [ ˆY,[ ˆX, ˆY]] = [ −it ℏmgˆz,igt2 ℏ ˆp ] = img2t3 ℏ .(B4) 14 Since[ ˆY,[ ˆX, ˆY]]is proportional to the identity, all higher nested c...

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QFIM entries for the centered Gaussian Take the centered minimum-uncertainty Gaussian state with Varψ0(ˆz) =σ2 2 ,Var ψ0(ˆp) = ℏ2 2σ2,Cov sym ψ0 (ˆz,ˆp) = 0.(B9) The QFIM entries areFij = 4 Covsym ψ0 (ˆHi, ˆHj ) . From Eq. (B7), the gravity generator is ˆHg = ˆG0(t) = t ℏ ( mˆz+t 2 ˆp ) ,(B10) so Fgg = 4 Varψ0 (t ℏ ( mˆz+t 2 ˆp )) = 4t2 ℏ2 ( m2 Varψ0(ˆz) ...

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The total Hilbert space is Hmot⊗C2

Internal KC fringe from pulse algebra We consider the symmetricπ/2–π–π/2sequence at pulse times t1 = 0, t 2 =T, t 3 = 2T.(D1) The motional Hilbert space is denoted byH mot and the internal basis by{|a⟩,|b⟩}. The total Hilbert space is Hmot⊗C2. Define the phase operators Θj :=k 0ˆz(tj)−ϕj, E j :=e +iΘ j, E † j =e−iΘj,(D2) whereˆz(tj)is the Heisenberg posit...

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+E† 2·1 E2·1 + 0·(−iE1)E 2(−iE†

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(D6) Now multiply byU3 from the left

+ 0·1 ) = −i√ 2 ( −iE† 2E1 E† 2 E2 −iE2E† 1 ) = 1√ 2 ( −E† 2E1 −iE† 2 −iE2 −E2E† 1 ) . (D6) Now multiply byU3 from the left. The amplitude to exit in|b⟩starting from|a⟩is the(b,a)entry ofU3U2U1. Since rowbofU 3 is 1√ 2 (−iE3,1),(D7) and columnaofU 2U1 is 1√ 2 (−E† 2E1 −iE2 ) ,(D8) their product gives ˆAb := (U3U2U1)ba = 1 2 [ (−iE3)(−E† 2E1) + 1·(−iE2) ] ...

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Differential KC benchmark: additive QFI and motional timing contribution We now derive the identities used in the full-state benchmark of Sec. VB. Under ideal closure, the final state can be written as |Ψ(g,T)⟩=|ψout(T)⟩⊗|χ(g,T)⟩,(D18) where|ψout(T)⟩is a motional state independent of the differential gravity phase and |χ(g,T)⟩=1√ 2 ( |a⟩+ei∆Φ(g,T) |b⟩ ) ....

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de Angelis, A

M. de Angelis, A. Bertoldi, L. Cacciapuoti, A. Giorgini, G. Lamporesi, M. Prevedelli, G. Saccorotti, F. Sor- rentino, and G. M. Tino, Measurement Science and Tech- nology20, 022001 (2009)

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Peters, K

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F. Armata, L. Latmiral, A. D. K. Plato, and M. S. Kim, Physical Review A96, 043824 (2017)

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S.Qvarfort, A.Serafini, P.F.Barker,andS.Bose,Nature Communications9, 3690 (2018)

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K. Bongs, M. Holynski, J. Vovrosh, P. Bouyer, G. Con- don, E.Rasel, C.Schubert, W.P.Schleich,andA.Roura, Nature Reviews Physics1, 731 (2019)

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Kasevich and S

M. Kasevich and S. Chu, Applied Physics B54, 321 (1992)

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S. S. Wani, S. Al-Kuwari, A. Shabir, P. Vezio, F. Marino, and M. Faizal, Phys. Rev. A113, 032609 (2026)

work page 2026

[11] [11]

Song and Q.-y

C.-G. Song and Q.-y. Cai, EPJ Quantum Technology12, 73 (2025)

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Herb and C

K. Herb and C. L. Degen, Physical Review Letters133, 210802 (2024)

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Kasevich and S

M. Kasevich and S. Chu, Physical Review Letters67, 181 (1991)

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M. Szczykulska, T. Baumgratz, and A. Datta, Advances in Physics: X1, 621 (2016)

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J. Rubio and J. Dunningham, Physical Review A101, 032114 (2020)

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Herbst, T

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S. Pang and T. A. Brun, Physical Review A90, 022117 (2014)

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J. S. Sidhu and P. Kok, AVS Quantum Science2, 014701 (2020)

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Blanes, F

S. Blanes, F. Casas, J. A. Oteo, and J. Ros, Physics Re- ports470, 151 (2009)

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Pelluet, R

C. Pelluet, R. Arguel, M. Rabault, V. Jarlaud, C. Mé- tayer, B. Barrett, P. Bouyer, and B. Battelier, Nature Communications16, 4812 (2025)

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Beaufils, L

Q. Beaufils, L. A. Sidorenkov, P. Lebegue, B. Venon, D. Holleville, L. Volodimer, M. Lours, J. Junca, X. Zou, A. Bertoldi, M. Prevedelli, D. O. Sabulsky, P. Bouyer, A. Landragin, B. Canuel, and R. Geiger, Scientific Re- ports12, 19000 (2022)

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Freier, M

C. Freier, M. Hauth, V. Schkolnik, B. Leykauf, M. Schilling, H. Wziontek, H.-G. Scherneck, J. Müller, andA.Peters,JournalofPhysics: ConferenceSeries723, 012050 (2016)

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Rodzinka, E

T. Rodzinka, E. Dionis, L. Calmels, S. Beldjoudi, A. Béguin, D. Guéry-Odelin, B. Allard, D. Sugny, and A. Gauguet, Nature Communications15, 10281 (2024)

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[31] [31]

L. A. Fuderer, J. J. Hope, and S. A. Haine, Physical Review A108, 043722 (2023). 12 Appendix A: Affine gravity–time cross term under the weak-commutator condition This appendix proves the structural statement used in Sec. III. Assume theg-independent dynamics satisfy the weak-commutator condition [ˆz0(u),ˆz0(v)] =iℏ∆(u,v)I, u,v∈[0,t],(A1) with∆(u,v)∈R. Th...

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[32] [32]

Exact BCH factorization Write ˆU(g,t) =e ˆX+ ˆY, ˆX:=−it ℏ ˆp2 2m, ˆY:=−it ℏmgˆz.(B2) Using[ˆz,ˆp] =iℏ, we first compute[ˆp2,ˆz] = ˆp[ˆp,ˆz] + [ˆp,ˆz]ˆp=−2iℏˆp. Hence [ ˆX, ˆY] = [ −it ℏ ˆp2 2m,−it ℏmgˆz ] = igt2 ℏ ˆp,(B3) [ ˆY,[ ˆX, ˆY]] = [ −it ℏmgˆz,igt2 ℏ ˆp ] = img2t3 ℏ .(B4) 14 Since[ ˆY,[ ˆX, ˆY]]is proportional to the identity, all higher nested c...

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[33] [33]

QFIM entries for the centered Gaussian Take the centered minimum-uncertainty Gaussian state with Varψ0(ˆz) =σ2 2 ,Var ψ0(ˆp) = ℏ2 2σ2,Cov sym ψ0 (ˆz,ˆp) = 0.(B9) The QFIM entries areFij = 4 Covsym ψ0 (ˆHi, ˆHj ) . From Eq. (B7), the gravity generator is ˆHg = ˆG0(t) = t ℏ ( mˆz+t 2 ˆp ) ,(B10) so Fgg = 4 Varψ0 (t ℏ ( mˆz+t 2 ˆp )) = 4t2 ℏ2 ( m2 Varψ0(ˆz) ...

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[34] [34]

The total Hilbert space is Hmot⊗C2

Internal KC fringe from pulse algebra We consider the symmetricπ/2–π–π/2sequence at pulse times t1 = 0, t 2 =T, t 3 = 2T.(D1) The motional Hilbert space is denoted byH mot and the internal basis by{|a⟩,|b⟩}. The total Hilbert space is Hmot⊗C2. Define the phase operators Θj :=k 0ˆz(tj)−ϕj, E j :=e +iΘ j, E † j =e−iΘj,(D2) whereˆz(tj)is the Heisenberg posit...

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[35] [35]

+E† 2·1 E2·1 + 0·(−iE1)E 2(−iE†

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[36] [36]

(D6) Now multiply byU3 from the left

+ 0·1 ) = −i√ 2 ( −iE† 2E1 E† 2 E2 −iE2E† 1 ) = 1√ 2 ( −E† 2E1 −iE† 2 −iE2 −E2E† 1 ) . (D6) Now multiply byU3 from the left. The amplitude to exit in|b⟩starting from|a⟩is the(b,a)entry ofU3U2U1. Since rowbofU 3 is 1√ 2 (−iE3,1),(D7) and columnaofU 2U1 is 1√ 2 (−E† 2E1 −iE2 ) ,(D8) their product gives ˆAb := (U3U2U1)ba = 1 2 [ (−iE3)(−E† 2E1) + 1·(−iE2) ] ...

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[37] [37]

Differential KC benchmark: additive QFI and motional timing contribution We now derive the identities used in the full-state benchmark of Sec. VB. Under ideal closure, the final state can be written as |Ψ(g,T)⟩=|ψout(T)⟩⊗|χ(g,T)⟩,(D18) where|ψout(T)⟩is a motional state independent of the differential gravity phase and |χ(g,T)⟩=1√ 2 ( |a⟩+ei∆Φ(g,T) |b⟩ ) ....

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