Sensitivity of a collisional single-atom spin probe

Referee: One of the main findings in Ref. 15 was that “the sensitivity can be significantly enhanced by considering nonequilibrium spin dynamics of the quantum probe.” Hence, it is not clear what the relevance of the current study, where a more detailed analysis of the steady state situation is presented. The steady state situation not only performs worse in terms of sensitivity (Fisher information) from a theoretical point of view, but is also of limited interested from an experimental point of view, because “for many combinations of thermal and Zeeman energies, the life time of the single-atom probe is too low to experimentally reach the steady state.”
In the introduction, after the authors presented their former findings in Ref. 15, it is stated “This raises the question if the performance and specifically the sensitivity of such quantum probes can be understood from the microscopic interaction mechanisms determining individual atomic collisions. Such understanding could open the door to further optimization of the probing process.” It is unclear to me whether this defines the aim and original contribution of the present paper. I would say that there is no new physical insight presented in this manuscript that was not already contained in Ref. 15

Authors: In ref. 15, the system is applied as a thermometer and magnetometer, focusing on the sensitivity in the short-time nonequilibrium regime.
There, however, it remained unclear how the sensitivity is connected to the microscopic collisional mechanisms and their rates. Moreover, new data presented in this work show that there are specific points in parameter space for B and T (not time), where the sensitivity is maximized. This is surprising at first glance because the collision rates all scale monotonically with magnetic field or temperature. We emphasize that these maxima occurring in (B, T) space for the infinite time limit are different from the nonequilibrium maxima of Ref. 15, which occur for constant (T, B) but varying time.
Here, we use equilibrium-state simulations to reveal the origin of these maxima in parameter space, which are related to the inflection points of the thermal energy distribution governing the endothermal collision probability.
Furthermore, we compare these equilibrium simulations to our nonequilibrium data to show that the positions are still well explained, while the quantitative difference depends on the equilibrium-nonequilibrium difference. We emphasize that simulations of rate equations would still yield very good agreement but do not offer additional insight into the origin of the sensitivity maxima in parameter space.
Our work, therefore, allows optimizing the operation of a quantum probe not only in time but additionally in the (B, T) parameter space.

We have clearly indicated the novel aspects and the differences to Ref. [15] in the revised version.

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Referee: The comparison with the experimental data (presumably reanalyzed from Ref.15) is rather hasty. The reader is left alone to interpret the experimental data in Fig. 7 compared to the theoretical curves, which could otherwise be the main result of this work. Thus, it is not clear what the main message is. If anything, the message the reader can take from this section is the opposite of what the authors may have desired, which is that steady state analysis is not helpful in describing a non-equilibrium situation.

Authors: It is not the goal of this paper to establish the best agreement between our measurements and the rate model; this agreement was established in Ref. [15], and it holds to a percent level for all parameter ranges considered.
By contrast, our steady-state model offers an intuitive explanation for the newly observed sensitivity maxima in parameter space and the connection between the microscopic collision mechanism and the sensitivity. The comparison to the nonequilibrium data is supposed to show that, despite quantitative discrepancies, the position of the maxima is well explained throughout the parameter space, and thus the microscopic explanation and intuition hold even out of equilibrium.
We have clarified in the manuscript that the rate model in principle can provide agreement, but that the purpose of the comparison is to compare the positions of the maxima rather than the absolute values.

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Referee: I believe that the authors' analysis deserves publication after due consideration of the criticisms. I hesitate however to recommend this work for publication in “SciPost Physics.” After considering the 4 acceptance criteria for the journal, at least one of which must be met, I could not find any that fits the present work.

Authors: We believe that, in view of our modifications in the revised version, the manuscript meets acceptance criterion 3 (Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work).
While the work in Ref. [15] indicated that the quantum probe operation can be optimized in time, the present work shows that optimization in parameter space (B,T) or, equivalently, total energy and energy ratio, independent of time is additionally available. This opens the door for dynamical optimization strategies of quantum probe operation, bringing together optimization and possibly machine learning with quantum probing. We believe that this is a fruitful ground for an experimental platform which will stimulate numerous future research directions, enabled by the observation and understanding of the sensitivity behavior in parameter space.
Moreover, the connection between experimental access to macroscopic and information theoretical quantities on the one hand, and the ability to see the impact of individual collisions on single atoms on the other hand, paves the way to experimentally address fundamental questions of quantum probing, such as the minimal perturbation necessary for a given sensitivity. Addressing such questions experimentally requires a thorough understanding of the system in the full parameter range considered, which we provide by our work. We emphasize that this is not only quantitative but qualitative progress in understanding the performance of the quantum probe, which can be employed for the new applications mentioned above.

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Referee: I have collected a series of technical remarks that I will list below and I would like to ask the authors to consider. In order of appearance in the manuscript
Technical remarks:
On a conceptual level, I find it misleading to speak of quantum Fisher information since their expression of the sensitivity function reduces to the classical Fisher information. The reason is that coherences play no role in the steady state that is considered in this work. In this situation, the density operators for different parameters commute and one can directly use the classical Hellinger distance instead the quantum Bures distance. Thus, one can obtain the same results presented in this work based on classical arguments only, without invoking any (in this case, not necessary and thus misleading) “quantumness”.
By working out the expression in Eq. (8), one finds the classical expression of the Fisher information:
dB2 (δθ) = 1/2 Fclassical δθ2
The physical reason is that the authors are not varying the POVM set to optimize the Fisher information, but use instead projectors on the basis of eigenstates of the density operator. The authors’ choice makes perfect sense under the situation analyzed, because coherences are suppressed in the steady state. However, there is no need to call the Fisher information “quantum.”
Perhaps that authors find it useful to consider W. K. Wootters, “Statistical distance and Hilbert space,” Phys. Rev. D 23, 357 (1981) where the notion of distance for classical distribution is defined (this was long known since Hellinger, 1909), and the link between quantum and classical is also established. Moreover, the connection between the Hellinger distance and the classical Fisher information is presented in Eq. (7.2.9) of "Geometrical Foundations of Asymptotic Inference" by Robert E. Kass, Paul W. Vos (John Wiley & Sons, Inc., 1997).
In Eq. (9), there is a factor 1/2 on the rhs that is missing, at least if the authors use the standard definition of the Fisher information. If they use a different definition, some reader might derive a wrong result for the minimum variance when applying the Cramér-Rao bound, because then a factor 2 would be missing.
I noticed that the factor 2 is also missing in Ref. 15.

Authors: We agree with the referee that in our case, the Fisher information does not show any signatures of quantum effects; we have changed “quantum Fisher information” into “Fisher information”, and we have added a note that for our case the Bures distance coincides with the classical Hellinger length.
We thank the Referee for pointing out the inconsistency between statistical speed and Fisher information. We agree that our definition of the Fisher information via the first derivative differs eventually from the conventional definition. We have therefore replaced eq. (9) by the definition of the statistical speed in terms of Bures distance and Fisher information. We prefer the statistical speed, i.e., the linear change, to indicate the validity range of this approximation. We have moreover clarified the connection between the statistical speed and the Fisher information.

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Referee:The line after Eq. (9) must contain a typo, where δ θ should rather be δθ .

Authors: 3) Equation (9), where δθ is considered, replaces this connection.

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Referee: In the figure 3(c), one of the processes is indicated as from mF = −1 to mF = 1 , where the mF changes by +2. This must be a typo since only ±1 first-order changes are considered.

Authors: 4) We have corrected the typo in the legend in Fig. 3 c.

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Refree: The units of the y-axis of Fig.4 are not clear. What does dB (delta B) for panel (a) means? I have similar questions for the other panels. I expect that the Bures distance is dimensionless, while it seems not to be the case from the label in the figure.

Authors: 5) We have removed the expression in the round brackets because, indeed, the Bures distance is dimensionless.

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Referee: Regarding the sentence “The obvious asymmetry of the Bures distance dBures with respect to the reference point results from an asymmetric energy condition for endoergic collision mechanisms and the differing collision rates of endo- and exoergic processes.” it is not sufficiently clear that the difference affects also the infinitesimal variations. The result is that one obtains two different Fisher information depending whether the infinitesimal variation is positive or negative.
I admit that I do not understand deeply why this is the case. I don’t see exactly the connection between this fact and the differing collision rates of endo- and exoergic processes. However, my more serious concern is that the entire concept of Fisher information and Cramér-Rao bound on the variance is not well defined, if the Hellinger (and Bures) distance squared does not define a differential form. Derivations of the Cramér-Rao bound rely on the differentiability of the Fisher information.
I understand that differences between the two slopes are small, of order of 20%. Empirically, one can use the average value, but then how does the Cramér-Rao bound get modified? Either the authors derive a mathematical foundation of the Cramér-Rao bound in this case, or it is clarified for the reader that said bound is considered on a heuristic basis (without any proof).

Authors: 6) We thank the referee for pointing out this question. We have tacitly taken the heuristic approach, assuming that Fisher information is defined because the opposite would mean that we have experimentally revealed a system where the standard concepts of information theory cannot be applied straightforwardly. Experimentally, our data and the intuitive picture emerging from our work show that the statistical speed is different for two opposite directions in parameter space. From our experimental point of view, this suggests that different sensitivities might be reasonable.

Moreover, it is certainly not the goal of the manuscript to theoretically justify or even prove the existence of a Fisher information or Cramér-Rao bound in this case. We have clarified that we use the heuristic approach.
Motivated by the numerically observed linear scaling around the zero point of the Bures distance, a first-order Taylor expansion is used as in [M. Gessner and A. Smerzi, Statistical speed of quantum states: Generalized quantum fisher information and schatten speed, Phys. Rev. A 97, 022109 (2018)].
We appreciate, however, that the notion of two Fisher information depending on the “direction” in parameter space is a case that might be interesting for future research, particularly because it is an experimentally relevant case. We note that this might stimulate relevant future theoretical work beyond the directions opened by our work mentioned above. We have added this point to the conclusions.

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Referee: In the y-label of Fig.5 and Fig.7 (b-g), I don’t understand the meaning of the sqrt of the infinitesimal variation. I think this should not be there. The units seem to be correctly chosen in Ref. 15 of the same authors, but not here.
Moreover, concerning Fig. 5, I don't understand why in (a) and (b) the B field and temperature are scanned, whereas in (c) and (d) an infinitesimal variation of the total energy and energy ratio are varied. Can there be a typo in the label of (a) and (b)?

Authors: 7) We changed the y-axis units of Fig. 5&7 and the x-axis in Fig. 5 (a, b).

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Referee: For Fig.6 I have similar questions as for Fig.5. Why is there an infinitesimal variation in the x-label, indicated by δE ratio ? This must probably be a typo. I would say that one is only interested in infinitesimal variations, with the caveat that in an experiment the variation cannot be infinitesimal but only suffciently small. I also remark here in passing that about the size of the experimental variations, I could not find information.

Authors: 8) We furthermore have clarified more explicitly the difference between Fig. 4 on the one hand and Figs. 5, 6, and 7, on the other hand. In Fig. 4, we compute the change of population distribution for infinitesimal deviation from the reference point. Here, we agree that strictly only infinitesimal changes should be considered. Our results show, however, that numerically within a significant range, the response is linear. For the (B,T) axes, this amount to few 10 mG and 100…200 nK. As Fig. 7 shows, this is of the order of the spacing of the experimental data. We note that much finer sampling, at least in the direction of temperature, is experimentally not reliably possible, which directly also affects the uncertainties in the parameter directions of total energy and energy ratio.
By contrast, Figs. 5, 6, and 7 display the response of Fisher information or probability distribution as a function of reference point. This reference point, however, can be chosen arbitrarily; for the numerical calculations, we compute the response to infinitesimal deviations from this reference point; for the experimental data, we compare different experimentally recorded populations and consider each data point in Fig. 7(a) that belongs to a line (indicated by the same symbols and colors) as a reference point. Interestingly, even for data points that show no infinitesimal deviations from the reference point and deviate from numerically investigated lines, the system displays a similar behavior as for the numerically investigated case.

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Referee: Regarding the sentence “As shown there, the sensitivity based on the nonequilibrium spin dynamics is significantly larger than the one using the steady state.” it is not fully clear to me whether the curves in Fig.7 are computed for the steady state situation? I presume that the answer is yes, though a clearer statement would be helpful.

Authors: 9) Curves in Fig. 7 are computed for the steady state situation. We have clarified this in the text. We emphasize again that this comparison serves to show that the steady state intuition still holds out of equilibrium for the positions of the sensitivity maxima. We have clarified this in the manuscript.

Referee: No clear statement on what is new compared to previous publications, in particular ref. 15.
What I miss is a clear statement on what is new with respect to the previous publications of the experimental group on the very topic, see in particular the refs. 15 and 19.
Please put a corresponding statement at the beginning on what is new here, please see my report.

Authors: Ref. [19] explored the atomic physics details of interspecies spin exchange and the consequences of, e.g., the angular momentum structure in this type of collisions. By contrast, the application of SE processes for sensing was not investigated, and thus the observation of sensitivity maxima in parameter space was elusive. This observation and its intuitive explanation are the focus of our actual manuscript.
In ref. 15, the system is applied as a thermometer and magnetometer, focusing on the sensitivity in the short-time nonequilibrium regime.
There, however, it remained unclear how the sensitivity is connected to the microscopic collisional mechanisms and their rates. Moreover, new data presented in this work show that there are specific points in parameter space for B and T (not time), where the sensitivity is maximized. This is surprising at first glance because the collision rates all scale monotonically with magnetic field or temperature. We emphasize that these maxima occurring in (B, T) space for the infinite time limit are different from the nonequilibrium maxima of Ref. 15, which occur for constant (T, B) but varying time.
Here, we use equilibrium-state simulations to reveal the origin of these maxima in parameter space, which are related to the inflection points of the thermal energy distribution governing the endothermal collision probability.
Furthermore, we compare these equilibrium simulations to our nonequilibrium data to show that the positions are still well explained, while the quantitative difference depends on the equilibrium-nonequilibrium difference. We emphasize that simulations of rate equations would still yield very good agreement but do not offer additional insight into the origin of the sensitivity maxima in parameter space.
Our work, therefore, allows optimizing the operation of a quantum probe not only in time but additionally in the (B, T) parameter space.

We have clearly indicated the novel aspects and the differences to Ref. [15] in the revised version.

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Referee: The correspondence between the model prediction and the theory is not really perfect and might be comment on further, see Fig. 7

Authors: It is not the goal of this paper to establish the best agreement between our measurements and the rate model; this agreement was established in Ref. [15], and it holds to a percent level for all parameter ranges considered.
By contrast, our steady-state model offers an intuitive explanation for the newly observed sensitivity maxima in parameter space and the connection between the microscopic collision mechanism and the sensitivity. The comparison to the nonequilibrium data is supposed to show that, despite quantitative discrepancies, the position of the maxima is well explained throughout the parameter space. Thus the microscopic explanation and intuition hold even out of equilibrium.
We have clarified in the manuscript that the rate model, in principle, can provide agreement but that the purpose of the comparison is to compare the positions of the maxima rather than the absolute values.

Referee: The main findings of the paper are not significantly different with respect to what the authors already published.
I nevertheless found that most of the main findings were already contained in the original publication in PRX by the same group (reference 15).

Authors: In ref. 15, the system is applied as a thermometer and magnetometer, focusing on the sensitivity in the short-time nonequilibrium regime.
There, however, it remained unclear how the sensitivity is connected to the microscopic collisional mechanisms and their rates. Moreover, new data presented in this work show that there are specific points in parameter space for B and T (not time), where the sensitivity is maximized. This is surprising at first glance because the collision rates all scale monotonically with magnetic field or temperature. We emphasize that these maxima occurring in (B, T) space for the infinite time limit are different from the nonequilibrium maxima of Ref. 15, which occur for constant (T, B) but varying time.
Here, we use equilibrium-state simulations to reveal the origin of these maxima in parameter space, which are related to the inflection points of the thermal energy distribution governing the endothermal collision probability.
Furthermore, we compare these equilibrium simulations to our nonequilibrium data to show that the positions are still well explained, while the quantitative difference depends on the equilibrium-nonequilibrium difference. We emphasize that simulations of rate equations would still yield very good agreement but do not offer additional insight into the origin of the sensitivity maxima in parameter space.
Our work, therefore, allows optimizing the operation of a quantum probe not only in time but additionally in the (B, T) parameter space.

We have clearly indicated the novel aspects and the differences to Ref. [15] in the revised version.

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Referee: The comparison to experiments is only partially relevant, due to signifcant differences between theory and experiment.
A careful analysis, providing better agreement and understanding of the data is desirable of course, but the agreement to experimental data seems to be overall poorer, due to the assumption of steady state operation, which is not realized in the experimental situation.

Authors: It is not the goal of this paper to establish the best agreement between our measurements and the rate model; this agreement was established in Ref. [15], and it holds to a percent level for all parameter ranges considered.
By contrast, our steady-state model offers an intuitive explanation for the newly observed sensitivity maxima in parameter space and the connection between the microscopic collision mechanism and the sensitivity. The comparison to the nonequilibrium data is supposed to show that, despite quantitative discrepancies, the position of the maxima is well explained throughout the parameter space. Thus the microscopic explanation and intuition hold even out of equilibrium.
We have clarified in the manuscript that the rate model, in principle, can provide agreement but that the purpose of the comparison is to compare the positions of the maxima rather than the absolute values.

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Referee: The conclusions are not properly written. They represent a list of main points to elaborate further.
I also found serious issues in the present conclusions, which appear to mostly be a list of statements lacking connections, with an overall lower language quality with respect to the rest of the paper.

Authors: We have rewritten the conclusion and emphasized the future potential of single atom probing, such as optimizing probing parameters during a measurement.