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Quantum sensing of a quantum field
by Ricard Ravell Rodríguez, Martí Perarnau-Llobet, Pavel Sekatski
Submission summary
| Authors (as registered SciPost users): | Ricard Ravell Rodríguez |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2509.22361v1 (pdf) |
| Date submitted: | Sept. 29, 2025, 10:33 a.m. |
| Submitted by: | Ricard Ravell Rodríguez |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
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 limit, where the atom interacts with many weak coherent states, this back-action can be simply interpreted as spontaneous emission, giving rise to the optimal interaction time and QFI rate.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
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In this manuscript the authors consider the problem of sensing the strength of a quantum field by probing it with a qubit in different scenarios. This is a problem relevant in many areas of quantum metrology. Moreover, they put emphasis on the important differences that appear when treating the field quantum-mechanically instead of as a parameter of the qubit Hamiltonian (semi-classical Rabi model).
The authors consider several quite different scenarios, which are: 1.) A qubit interacting with a single mode field. 2.) A qubit interacting sequentially with many modes in different limits.
In all cases the modes are set into a coherent state, and the goal is to estimate the strength of the field.
These situations are physically quite different. Nevertheless, the authors manage to treat them in a systematic way: in all cases the basis of the treatment is Eq. (19) and its subsequent solution/approximation in different limits. They also show that a simple upper bound obtained from analyzing the QFI of coherent states [Eq. (4)] is useful in all cases.
I think that this unified treatment makes the work quite original and interesting. Furthermore, the developed solutions for the reduced atom dynamics and QFI might be interesting beyond the field of quantum sensing, or also inspire similar treatments for different situations of interest in the field.
I think that the manuscript meets the expectations stated for SciPost, and generally meets the acceptance criteria. However, there are some points that I would recommend to address before publication, regarding clarity of the presentation and discussion of the results in a broader context.
In particular:
Regarding additional discussion:
1.) The authors analyze quite different scenarios, single-mode vs. many modes in different limits, and also compare them with the semi-classical Rabi model. I think the manuscript would benefit from a more extended discussion about which are the physical situations/platforms in which each of the models is expected to be relevant.
2.) The general bound of Eq. (4) and its application to the last scenario (Sec. IV C), i.e. Eq. (90), reminded me of an apparently similar result presented in some works by AH Kiilerich and K Mølmer:
AH Kiilerich, K Mølmer, Phys. Rev. A 89, 052110 (2014); Phys. Rev. A 94, 032103 (2016).
Both consider estimation of the Rabi frequency of a driven qubit with spontaneous emission. In the first case, the sensing protocol is based on photon counting while in the second is based on homodyne monitoring. They show that with these methods they can achieve a QFI bound that is given by $4T/\Gamma$, where $\Gamma$ is the emission rate and $T$ the duration of the protocol [see e.g. Eq. (14) of the arxiv version of the first work].
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First, can you comment on how is their QFI bound related to your Eqs. (4) and (90)?
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Second, wouldn't the protocols of these works answer the question of which is the optimal way to probe the field in the scenario of Sec. IV C?
3.) Some points regarding the presentation:
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In Fig. 1 (and 2 to a lesser extent) the lines on top of the heat maps are difficult to visualize. Especially the red and blue solid lines. I think this should be improved.
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Fig. 2 bottom panels. What is the meaning of the white patches appearing for small coherent amplitudes? This should be clarified.
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There seems to be a typo in Eq. (33). In the left-hand side the sub-index "r" is used, but this symbol does not appear in the right-hand side.
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Page 7: "Notice that the channel is not fully entanglement-breaking as long as the dephasing is not complete, r > 0." It is not clear to me what "r" stands for.
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Page 11. There seems to be a couple of typos. First, in the definition "N=1/dt". Second, shouldn't the exponent in Eq. (78) be "gt dt"?
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Page 17. "Before discussing the case of coherent field, let us say a few words about the reduced atomic given by the channel E in Eq. (19) in general." This sentence looks confusing.
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Page 19. It is not clear to me what are the coefficients $c(k,\pm)$, what is their significance and how are they calculated. Perhaps this should be stated more clearly.
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Page 19. Is it possible that you mean that $\Gamma_{ij}(n)=1+O(\alpha^{-1})$ instead of $\Gamma_{ij}(n)=O(\alpha^{-1})$?
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Page 19. Possible typo: "More precises expanding"
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When the field is treated classically it is known that the QFI scales quadratically with the interaction time, therefore it is not bounded from above. Then the field is treated quantum mechanically and the resonant Jaynes-Cummings model is considered. It is found that when the interaction is with only one mode (in a coherent state) then the QFI is upper bounded by 4 which can only be achieved for very low amplitude. For a larger field amplitude its maximal value is 1.47. Then, the authors consider the interaction with a sequence of N coherent states of the same amplitude where the optimum 1.47*N can be achieved for short interaction times. Finally, the authors also take the continuous limit, where N goes to infinity and they calculate the optimal QFI.
The topic is timely and interesting as it combines concepts from quantum metrology and models from quantum optics. It characterizes how precisely one can measure the field amplitude via the interaction with a two-level system.
I think the manuscript is of very high quality, logically structured thus it is easy to follow. The derivations both in the main text and in the appendices seem plausible, correct and reproducible.
I recommend its publication in SciPost Physics. I have the following minor comments and questions:
• Is the RWA assumed for Eq. (1)? In my opinion it should be stated even if it is standard.
• Already after Eq. (3) it would be good to hint some intuitive arguments why we cannot obtain this scaling for the more realistic scenarios. In this regard, probably it could also be useful for the reader to discuss briefly why can Eq. (3) go to infinity.
• I am wondering if there are previous works considering the QFI in a quantum optical setting. If yes, they should be cited in the Introduction. Altogether, expanding the Introduction a little bit would be great (e.g., by hinting some of the results intuitively already at this point)
• Could the authors comment on possible experimental demonstration of their schemes?
• After Eq. (11) the authors say that “Note that the expressions on the right-hand side appear to diverge in the limit where rho_theta is pure.” Can the authors state intuitively why is that? In quantum metrology when one considers an initial state psi that is pure and a Hamiltonian H that generates the unitary dynamics then the QFI is just the variance of H in the state psi. How does this divergence align with this picture?
• Is there an intuitive explanation why the SWAP unitary in Eq. (15) corresponds to the optimum for small alpha? And also, why does alpha^dot enter?
• I think in Figs 1,2 it would be great to have labels for the axes so that one does not have to infer it from the caption.
• In the text at the end of III. C the authors should emphasize more that they basically investigate two cases, alpha small and alpha large, and for the large alpha case they consider different interaction times.
• Also, in the text at the end of III. C, |x| refers to the coherence parameter from Fig. 1?
• The authors should emphasize more what is the advantage of considering a sequence of interactions already at the beginning of Section IV. Or just give there a motivation why do we consider this case apart from being a natural generalization.
• Is there an intuitive argument why can subsequent interaction help in the short time limit?
• Eq. (74) seems like we have the same maximum as before, which increased because we increased the number of resources (we have now N coherent states). So why can one consider this as an advantage compared to the non-sequential case?
• In Section IV. C, the authors should briefly explain intuitively the physical meaning of this type of limit.
• Why is the previously given bound for the QFI (i.e., 4) is now only valid for the QFIRate (in Section IV.)? Is it because the QFIRate is normalized by the number of modes used?
• Do I understand correctly that this scheme does not give a better precision than for example homodyne detection of the field, it could serve as an alternative method?
• (Just out of curiosity) What happens if one uses squeezed states? Could the QFI increase?
• (Just out of curiosity) Would considering a three level (with equally spaced levels) system help in increasing the QFI?
• (Just out of curiosity) Can the authors comment on how the results would change if there is some detuning (i.e. the Jaynes-Cummings model is non-resonant)?
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