Optimal control of a quantum sensor: A fast algorithm based on an analytic solution

We would like to thank the Referees for the thorough assessment of our work, which helped us to improve our manuscript.

We gratefully acknowledge that both Referees 1 and 2 support the publication of our work in SciPost Phys. We appreciate very much their opinion that ``this manuscript reports on a well thought-out and worthwhile study, with a clear transformative elements'', and that ``the line or argument is theoretically appealing'' and ``another great asset consists in the experimental verification''.

In the List of changes, we address point-by-point all the Referees' comments.

Response to Referee 1

\textbf{Referee 1. Requested change 1}: {\it It must be emphasized early on that $h(t)$ must be known beforehand. Only the amplitude can be detected. Please discuss whether this is a realistic situation occurring in practice! Me, being a theorist, am a bit skeptical whether the by far more relevant goal would not be to detect $b(t)$ including its time dependence.
Similarly, please state clearly, that the power spectrum $S(\omega)$
of the noise needs to be known a priori.}

\textbf{Response}: We thank the Referee for the comment. We added a phrase in the second paragraph to point out that our optimization method requires the previous knowledge of the target field time dependence and that of the noise spectrum. On the other hand, we introduce the fact that $h(t)$ must be known beforehand when this variable is first introduced, in Section 2. We now added a phrase to stress that knowing beforehand the time dependence of the target field has applications such as sensing spins ensembles or spins in molecules, or sensing weak signals (e.g. the action potential) coming from biological samples. Moreover, our optimization method can be part of a protocol where the time-dependence is first (sub-optimally) determined and then then the amplitude detected optimally.

\textbf{Referee 1. Requested change 2}: {\it Emphasize early on that you are dealing with *dephasing* only, omitting other decoherence processes. Justify that ``dephasing only'' is still a relevant issue.}

\textbf{Response}: We thank the Referee for pointing this out. In the abstract, Introduction and in Section 2 we state that the noise we are dealing with is dephasing noise. The main idea of implementing DD sequences is to overcome the loss of coherence due to dephasing noise [Barry et al., Rev. Mod. Phys. 92, 015004 (2020), Degen et al., Rev. Mod. Phys. 89, 035002 (2017)]. This is extremely relevant since most of the quantum sensing techniques are based on retrieving information stored in the coherent phase of the quantum sensor. Dephasing noise is also the strongest source of noise for NV-based sensors and many others.

\textbf{Referee 1. Requested change 3}: {\it What is meant by the "overlap" between noise and target field between Eq.~(3) and (4)? Please be a bit more precise.}

\textbf{Response}: We thank the Referee for the suggestion. What we mean is that simple signals with frequencies that are well off-resonance with the frequencies composing the noise can be already amplified by Carr-Purcell or generalized Carr-Purcell control signals. We modified the text accordingly.

\textbf{Referee 1. Requested change 4}: {\it The quantity $\eta$ is called "sensitivity", but later minimized. From the sense of the word, however, the sensitivity should be large - so I would rather call $1/\eta$ sensitivity.}

\textbf{Response}: We appreciate the Referee's suggestion. Our notation may be confusing since some times people refer to an improved sensitivity as ``high sensitivity'', however in literature it is very common to refer to the sensitivity as the minimum signal that can be measured [\href{https://doi.org/10.1103/RevModPhys.89.035002}{Degen et al., Rev. Mod. Phys. 89, 035002 (2017)}, \href{https://doi.org/10.1103/RevModPhys.92.015004}{Barry et al., Rev. Mod. Phys. 92, 015004 (2020)}, \href{https://www.frontiersin.org/articles/10.3389/fphy.2023.1212368/full}{Bai et al., Front. Phys. 11, 1212368 (2023)}]. It makes sense then to minimize it in order to improve the sensor capability to detect weak signals. In view of the above, we prefer to keep the current definition.

\textbf{Referee 1. Requested change 5}: {\it In Eq. (11) the inverse of a delta-function is used in the continuum. Please comment on what this really means and how one can define it mathematically.}

\textbf{Response}: We thank the Referee for the question. The delta function represents the kernel of the identity operator on the space of square-integrable functions. Its inverse is therefore another delta function. We modified the text explaining more carefully the meaning of the inverse.

\textbf{Referee 1. Requested change 6}: {\it After Eq. (15) please state that you take lambda to be constant. The notation $\lambda(t)=\lambda$ is not unambiguous.}

\textbf{Response}: We thank the Referee for the suggestion. We modified the text as requested.

\textbf{Referee 1. Requested change 7}: {\it Fig. 2 should be rendered much larger, including the fonts, to reach a decent readability.}

\textbf{Response}: We thank the Referee for the suggestion. We modified the figure as requested.

\textbf{Referee 1. Requested change 8}: {\it An important point needs to be elucidated in the experimental part: How can the "true" value of b read off and with which error? This must be elucidated, at best by a figure.}

\textbf{Response}: We agree with the referee that this point was not explicitly stated in our manuscript. For a fixed value of $T$, the value of $b$ is obtained by measuring the probability $P(T,b)$, i.e. the probability for the NV spin to stay in the initial state, estimated from the frequency of the $\left|0\right\rangle$ result in repeated measurements. The value of $b$ is then estimated from the measurement and the (known) functional dependence of $P(T,b)$ on $b$. The error of this estimate is bounded by the sensitivity. The sensitivity reported in Fig. 5a is optimized assuming to implement ``slope detection'' [see Degen et al., Quantum sensing, Rev. Mod. Phys., 89, 3 (2017) -- Sec. IV.E]. Therefore, to achieve this sensitivity it is necessary to be in the maximum slope section of the curves 5b. Therefore, to minimize the sensitivity, it is desirable to have the slope as steep as possible for the first half period of the oscillation shown in Fig.5b. In that range, there is an approximate linear relation between the variable to be measured $b$ and the output of our experiments $P(T,b)$, thus simplifying its estimate. In this sense, an improved sensitivity comes at the cost of a smaller dynamic range. Notice though that measuring fields higher than the ones in the first half period can be achieved by changing the range of the sensor (e.g.\ by changing the final observable--phase of the final $\pi/2$ pulse, a global phase will be added to the signal $P(T,b)$). A similar explanation was added in a new paragraph at the very end of Section 4. We hope that this explanation is enough, we believe that a figure is not necessary, since it won't be different from others in literature [see e.g., Fig. 3 in Ref. Degen et al., Quantum sensing, Rev. Mod. Phys., 89, 3 (2017)].

\textbf{Referee 1. Requested change 9}: {\it Fig. 6, panel b) the tick labels on the y-axis should start at 0.}

\textbf{Response}: We have re-scaled the figure to include 0 in the y-axis

\textbf{Referee 1. Requested change 10}: {\it Why does the rightmost panel of Fig. 7 not contain data for Sph.?}

\textbf{Response}: The case Sph. was not included in the right most panel of Fig. 7 because we didn't perform an experiment for that specific case. We could, however, include the simulated value (square) of the inverse sensitivity. A new version of the figure was implemented.

\textbf{Referee 1. Requested change 11}: {\it Finally, a careful reading by an English native speaker can improve the manuscript by leading to more idiomatic sentences at several occasions. "A minima" should read "A minimum", on page 9.}

\textbf{Response}: We thank the Referee for the suggestion. We improved the text following these suggestions and thoroughly spell-checked the manuscript.

Response to Referee 2

\textbf{Referee 2. Requested change 1}: {\it The overall clarity of the manuscript would be much increased if the abstract already specified the usage of the word "optimal", with a sentence to the effect of "in the sense that it optimises the sensitivity, i.e., the smallest detectable signal". It would also be beneficial to explicitly state in the introduction that the minimisation of the sensitivity is analogous to the classical Hamiltonian minimisation (upon time discretisation).}

\textbf{Response}: We appreciate the Referee for their suggestions. We have added a phrase in the abstract and at the beginning of the Introduction to improve the clarity of the manuscript. And we explicitly state in the introduction and in Fig 1 the connection between minimizing the sensitivity and finding the ground state of the classical Hamiltonian.

\textbf{Referee 2. Requested change 2}: {\it In deriving (A.4), why is it that C can be set to 1?}

\textbf{Response}: We thank the Referee for the question. In Eq.~(A.4) the constant $C$ can be set to 1 since it represents an overall scale factor in the detection efficiency, fixed by the experimental architecture~[Degen et al, Rev.\ Mod.\ Phys.\ 89, 035002 (2017)]. As such, it affects all the frequencies in the same fashion, and does not modify the optimization. We modified the text to highlight this fact.

\textbf{Referee 2. Requested change 3}: {\it In Eqs.(2,4), I take it Phi(T)=Phi(T,b)/b ? (Since Phi(T,b) is proportional to b?) This should be clarified.}

\textbf{Response}: We thank the Referee for the comment. Our notation was not clear, since $\varphi(T,b)$ was sometimes written as $\varphi(T)$. We purged the text from these inconsistencies.

\textbf{Referee 2. Requested change 4}: {\it Eq.(4): absolutely explain the operational significance of the sensitivity}

\textbf{Response}: We thank the referee for pointing out this. The sensitivity is the smallest detectable signal (variation) above the noise level. Operationally, in a magnetometer, $\eta$ is the minimum variation of the target field $b$ that can be measured in a slope detection protocol. We have now modified the paragraph around Eq.~(4) to include more details about the sensitivity in our experiments.