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arxiv: 2509.22361 · v2 · submitted 2025-09-26 · 🪐 quant-ph

Quantum sensing of a quantum field

Ricard Ravell Rodr\'iguez , Mart\'i Perarnau-Llobet , Pavel Sekatski This is my paper

Pith reviewed 2026-05-18 12:38 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum metrologyquantum Fisher informationcoherent statesJaynes-Cummings modelquantum sensingback-actionreduced density matrixspontaneous emission
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The pith

Estimating the amplitude of a coherent quantum field with a two-level atom produces a quantum Fisher information bounded by 4 due to coherent-state overlap.

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

The paper contrasts two versions of the same metrological task. In the familiar semi-classical Rabi model a two-level atom estimates the amplitude of a classical field and the quantum Fisher information grows quadratically with interaction time, independent of the amplitude value. In the fully quantum version the field is a single coherent state mode and the atom interacts through the Jaynes-Cummings Hamiltonian; after the field is traced out the atom’s reduced state yields a quantum Fisher information that cannot exceed 4. This constant upper bound arises because coherent states with different amplitudes are never fully orthogonal. The bound is reached only in the vacuum limit; for large mean photon number the peak value is instead 1.47 and occurs at interaction times of order 1 and order alpha squared, with later periodic revivals.

Core claim

When the atom interacts with a single coherent mode of the field, the QFI is bounded by 4, a constant dictated by the non-orthogonality of coherent states. We find that this bound can only be approached in the vacuum limit. In the limit of large amplitude alpha, the QFI is found to attain its maximal value 1.47 at tau = O(1) and tau = O(alpha squared), and also shows periodic revivals at much later times.

What carries the argument

Quantum Fisher information of the reduced density matrix of the two-level atom after tracing out the quantized coherent field under the Jaynes-Cummings interaction.

If this is right

  • For a sequence of coherent states the quantum Fisher information grows at most linearly with the number of modes because of entanglement created between atom and field.
  • In the continuous weak-coherent-state limit this back-action appears as spontaneous emission and the optimal information rate per unit time is finite and set by the source intensity.
  • The information rate extracted by the atom is always upper-bounded by the rate at which the radiation source itself emits quantum Fisher information.
  • Periodic revivals of the atomic QFI occur at later interaction times even for large-amplitude fields.

Where Pith is reading between the lines

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

  • The linear scaling bound for multiple modes suggests that any realistic continuous quantum field probe will ultimately be limited by cumulative entanglement with the environment.
  • Replacing the single atom with a collective spin or using different initial field states might relax the constant bound while preserving the coherent-state overlap mechanism.
  • The spontaneous-emission interpretation opens a direct link between quantum metrology and open-system master equations for continuous monitoring of quantum sources.

Load-bearing premise

All metrological information about the field amplitude is contained in the reduced state of the atom after the field degrees of freedom are discarded.

What would settle it

A calculation or experiment that extracts a quantum Fisher information larger than 4 from the atomic reduced state for a single non-vacuum coherent mode would falsify the claimed bound.

Figures

Figures reproduced from arXiv: 2509.22361 by Mart\'i Perarnau-Llobet, Pavel Sekatski, Ricard Ravell Rodr\'iguez.

Figure 1
Figure 1. Figure 1: FIG. 1. From top to bottom: the coherence [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. From top to bottom: the purity [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The QFI [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The function [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
read the original abstract

Estimating a classical parameter encoded in the Hamiltonian of a quantum probe is a fundamental and well-understood task in quantum metrology. A textbook example is the estimation of a classical field's amplitude using a two-level probe, as described by the semi-classical Rabi model. In this work, we explore the fully quantum analogue, where the amplitude of a coherent quantized field is estimated by letting it interact with a two-level atom. For both metrological scenarios, we focus on the quantum Fisher information (QFI) of the reduced state of the atomic probe. In the semi-classical Rabi model, the QFI is independent of the field amplitude and grows quadratically with the interaction time $\tau$. In contrast, when the atom interacts with a single coherent mode of the field, the QFI is bounded by 4, a constant dictated by the non-orthogonality of coherent states. We find that this bound can only be approached in the vacuum limit. In the limit of large amplitude $\alpha$, the QFI is found to attain its maximal value $1.47$ at $\tau =O(1)$ and $\tau =O(\alpha^2)$, and also shows periodic revivals at much later times. When the atom interacts with a sequence of coherent states, the QFI can increase with time but is bounded to scale linearly due to the production of entanglement between the atom and the radiation (back-action), except in the limit where the number of modes and their total energy diverge. Finally, in the continuous-field limit, where the atom interacts with a continuous source of weak coherent states, this back-action can be simply interpreted as spontaneous emission; we find that the optimal atomic QFI rate is finite, depends on the source intensity, and is upper bounded by the constant rate at which the QFI is emitted by the radiation source.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript compares quantum metrology of a field amplitude using a two-level atom in the semi-classical Rabi model versus the fully quantum case with a coherent-state field. In the semi-classical setting the QFI of the reduced atomic state grows quadratically with interaction time τ and is independent of amplitude. For a single coherent mode the QFI is bounded above by 4 (a consequence of the non-orthogonality of coherent states) and is approached only in the vacuum limit; for large α the maximum value attained is 1.47 at τ = O(1) and τ = O(α²) with later periodic revivals. Interaction with a sequence of coherent modes yields linear-in-time scaling except when both mode number and total energy diverge. In the continuous weak-coherent-state limit the back-action is interpreted as spontaneous emission and the optimal QFI rate is finite and intensity-dependent.

Significance. If the derivations hold, the work supplies a concrete illustration that the metrological advantage of a quantum probe is fundamentally limited by the non-classical character of the field and by the entanglement generated during the interaction. The explicit bounds, the vacuum-limit recovery of the semi-classical scaling, and the spontaneous-emission interpretation in the continuum limit are useful reference points for quantum-sensing protocols that employ quantized light.

major comments (2)
  1. [§3] §3, paragraph after Eq. (12): the claim that the QFI is bounded by 4 follows directly from the fact that the QFI of a coherent state |α⟩ equals 4 and is non-increasing under any CPTP map; the manuscript should state this general fact explicitly before the explicit Jaynes-Cummings calculation so that the bound is seen to be model-independent rather than an artifact of the Rabi Hamiltonian.
  2. [§4.2] §4.2, sentence containing the numerical value 1.47: the reported maximum for large α is obtained from numerical maximization of the QFI expression; the manuscript should supply the analytic expression for the reduced atomic density matrix (or at least the Bloch-vector components) used to obtain this number so that the result can be reproduced without re-deriving the full time-evolution operator.
minor comments (2)
  1. [Abstract] Abstract: the phrase “the QFI is bounded by 4, a constant dictated by the non-orthogonality of coherent states” is correct but would benefit from a parenthetical reference to the known QFI formula for coherent states.
  2. [Figure 2] Figure 2 caption: the time axis is labeled in units of the vacuum Rabi frequency; it should be stated whether the plotted curves are for fixed α or for the rescaled time τ/α.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment and constructive comments, which have helped us improve the clarity of the manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [§3] §3, paragraph after Eq. (12): the claim that the QFI is bounded by 4 follows directly from the fact that the QFI of a coherent state |α⟩ equals 4 and is non-increasing under any CPTP map; the manuscript should state this general fact explicitly before the explicit Jaynes-Cummings calculation so that the bound is seen to be model-independent rather than an artifact of the Rabi Hamiltonian.

    Authors: We agree with the referee that explicitly invoking the general monotonicity of the QFI under CPTP maps makes the origin of the bound clearer and independent of the specific interaction Hamiltonian. In the revised manuscript we have inserted a brief statement of this general fact immediately before the Jaynes-Cummings calculation, noting that the QFI of any coherent state |α⟩ is exactly 4 and cannot increase under the partial trace over the field. revision: yes

  2. Referee: [§4.2] §4.2, sentence containing the numerical value 1.47: the reported maximum for large α is obtained from numerical maximization of the QFI expression; the manuscript should supply the analytic expression for the reduced atomic density matrix (or at least the Bloch-vector components) used to obtain this number so that the result can be reproduced without re-deriving the full time-evolution operator.

    Authors: We thank the referee for this suggestion. The reduced atomic state in the Jaynes-Cummings interaction with a coherent field is obtained by tracing the unitary evolution over the field modes, yielding explicit (albeit lengthy) expressions for the Bloch-vector components in terms of the Poisson-weighted overlaps of displaced number states. In the revised manuscript we now provide these analytic expressions for the Bloch vector in §4.2, allowing direct numerical evaluation of the QFI without recomputing the full time-evolution operator. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from standard QM

full rationale

The paper computes QFI of the reduced atomic state via the standard Jaynes-Cummings/Rabi Hamiltonian acting on a coherent-state field, followed by partial trace. The bound of 4 follows directly from the known QFI of a coherent state |α⟩ (equal to 4 for real α) being non-increasing under any CPTP map, including the partial trace after a θ-independent unitary. All explicit calculations (vacuum limit, large-α value 1.47, revivals, multi-mode linear scaling, continuous limit as spontaneous emission) are performed from first-principles quantum mechanics and the overlap properties of coherent states. No fitted parameters are renamed as predictions, no self-definitional loops, and no load-bearing self-citations or ansatzes are invoked. The central claims remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard quantum mechanics and quantum optics without introducing new free parameters, ad-hoc entities, or unproven axioms beyond the usual coherent-state formalism and reduced-state QFI definition.

axioms (2)
  • standard math Coherent states are non-orthogonal with overlap determined by their amplitude difference
    Invoked to bound the QFI by 4; this is a textbook property of coherent states.
  • domain assumption The atom-field interaction is governed by the standard Rabi or Jaynes-Cummings Hamiltonian
    Used throughout the semi-classical and quantum comparisons (abstract).

pith-pipeline@v0.9.0 · 5876 in / 1658 out tokens · 62962 ms · 2026-05-18T12:38:14.958468+00:00 · methodology

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Reference graph

Works this paper leans on

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    √ ˆn α cos τ √ ˆn+ 1 sin τ √ ˆn # z(e) α = 1−2E h cos2(τ √ ˆn+ 1) i (C18) ˙x(e) α =−2E

    F. Schäfer, P. Sekatski, M. Koppenhöfer, C. Bruder, and M. Kloc,Control of stochastic quantum dynamics by dif- ferentiable programming,Machine Learning: Science and Technology2, 035004 (2021). 16 Appendix A: QFI of a two-dimensional system For a general two dimensional stateρθ = aθ bθ −ic θ bθ + icθ 1−a θ , and its derivative˙ρθ ≡ ∂ρθ ∂θ = ˙aθ ˙bθ −i ˙cθ ...

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    The linear phase regime When the interaction timeτis relatively slow, the phases are well approximated by linear functions on the support of the Poissonian distributions. More precises expanding inˆδ= ˆn−α2 α =O(1)one finds S0(n), S1(n), S2(n) = 2ατ+τ n−α 2 α +τ O(α −1)(D5) S3(n) =− τ 2α + τ 4α2 n−α 2 α +τ O(α −3).(D6) Using the moment generating function...

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    (D4), we follow the approach presented in [31]

    The rapid phases regime In the limit where the linear phase approximation fails, to compute such sums in the right-hand side of Eq. (D4), we follow the approach presented in [31]. For simplicity we now focus on the phases fork= 0,1,2and come back to the casek= 3at the end of the section. First, we use the Poissonian summation [48] to express the discrete ...

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    (D4) are given by ∞X n=0 P(n|α)Γ ij(n)eiSk(n) ≈    P∞ ν=0 I0,ν k= 0,1P∞ ν=0 I0,ν(−1)ν k= 2P∞ ν=0 I3,ν k= 3 ,(D24) with all the terms given in Eqs

    The full reduced dynamics of the atom Summarizing the above calculations, we find that in the limit of largeαwith τ/α3 →0and up toO(α −1)the expected values in Eq. (D4) are given by ∞X n=0 P(n|α)Γ ij(n)eiSk(n) ≈    P∞ ν=0 I0,ν k= 0,1P∞ ν=0 I0,ν(−1)ν k= 2P∞ ν=0 I3,ν k= 3 ,(D24) with all the terms given in Eqs. (D17, D8, D7). This allows us to express ...

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    The atomic state and its QFI To discuss the final state for an atom prepared in the ground or excited state and its QFI, we again separately consider the three different time regimes a. Quantum Fisher information at short timesτ=O(1) Forτ≪αusing Eq. (D31) one immediate obtainsx (g/e) τ|α =±e −τ 2/2 sin(2ατ),z (g/e) τ|α =±e −τ 2/2 cos(2ατ)(where the sign±i...

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    First, let us show that in this limit one recovers the atom coupled to a bosonic field models, studied in e.g

    1−τ 2(α2 + 1)−τ α τ α −τ α−τ α τ 2(α2 + 1)−τ 2α2 τ α τ α−τ 2α2 τ 2α2   =   1 1−αdt αdt 1 1−αdt αdt −αdt−αdt0 0 αdt αdt0 0   +O(dt 2)(F1) so the atomic dynamics is unitary d dt ρt|ε =−iα[σ y, ρt|ε].(F2) Appendix G: The continuous quantum field limit In this section, we compute the limit where the interaction parameter for a single discrete mode i...

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    Taking the field mode in the coherent state|α⟩= ε √ dt E , we further obtainG= ˜G+O((τ α) 3)with ˜G=   1−τ 2α2 1−τ 2(α2 + 1 2)−τ α τ α 1−τ 2(α2 + 1

    1−τ 2(⟨ˆn⟩+ 1)−τ a† τ⟨a⟩ −τ⟨a⟩ −τ⟨a⟩τ 2(⟨ˆn⟩+ 1)−τ 2 a†2 τ a† τ a† −τ 2 a2 τ 2 ⟨ˆn⟩   ,(G5) where the expected values⟨A⟩= trA|F⟩ ⟨F|are taken on the initial state of the incident radiation mode. Taking the field mode in the coherent state|α⟩= ε √ dt E , we further obtainG= ˜G+O((τ α) 3)with ˜G=   1−τ 2α2 1−τ 2(α2 + 1 2)−τ α τ α 1−τ 2(α2 + 1

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    1−τ 2(α2 + 1)−τ α τ α −τ α−τ α τ 2(α2 + 1)−τ 2α2 τ α τ α−τ 2α2 τ 2α2   =   1 1− κdt 2 −√κεdt √κεdt 1− κdt 2 1−κdt− √κεdt √κεdt −√κεdt− √κεdt κdt0√κεdt √κεdt0 0   ,(G6) where we only kept the leading orders indtin the last expression. It is then straightforward to verify that the resulting channel is of the form Edt|ε[•] = 3X i,j=0 Gji Li •L † ...

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