Noise Differentiation and Atom Number Measurement in Optical Lattice Clocks by Analyzing Clock Stabilities with Various Parameters

Round 1

Reviewer 1 Report

Dear Authors,

The article shows a method for evaluation number of atoms by extraction of noises in optical lattice clocks. It could be important for some case of use but in normal operation of optical clock the number of atoms it is not necessary. Moreover presented methods are only numerical although the group develops and operates optical clocks in the laboratory. Therefore there are direct possibilities to experimental proof your numerical concept. I understand that proposed theoretical work is essential but more relevant is to compare theory with experimental conditions. Firstly it means presented work it is not full study for the article. Secondly it means presented study does not match to Applied Science journal because there is not application of this research in the real experiment. I highly recommend to reconsider the article and improve the quality of this research by verifying presented numerical model with real data from normal operation of optical clock and then send as a new article.

The article is hard to read.

Author Response

First and foremost, we would like to express our gratitude to the referee for recognizing the potential significance of our work for practical applications. Precisely determining the number of atoms is crucial for studying many-body interactions and developing techniques to mitigate the density shift in optical lattice clocks. For instance, some of these techniques include the use of cold-collision-shift cancellation methods with a 'magic trap depth' (Alexander Aeppli, et al., Sci. Adv. 8, eadc9242 (2022) ) or at specific excitation fractions (A. D. Ludlow, et al., Phys. Rev. A. 84, 052724 (2011), Sangkyung Lee, et al., New J. Phys. 18, 033030 (2016)). Furthermore, accurate information about the atom number plays a pivotal role in exploring entangled states that can surpass the standard quantum limit and enhance the stability of optical clocks (William J. Eckner, et al., Nature, 621, 734 (2023)).

While the referee has raised concerns about the lack of experimental support for our numerical results, we have expanded our simulation process in the revised manuscript to make it easier to understand our numerical simulation method. We hope that these improvements can address referee’s concerns. Although we have the ability to perform experiments, obtaining the exact atom number (with uncertainty below 5%) is challenging, which limits the value of comparing numerical and experimental results. By contrast, numerical simulations provide more comprehensive information regarding measurement accuracy, uncertainty, atom number fluctuation, measurement time, and their effects on atom number measurements. Notably, numerical simulation enables us to fully examine the influence of modulation parameters α and β on measurement uncertainty.

Currently, some researches have inferred the standard quantum limit through in situ measurement-based differential clock comparisons (William J. Eckner, et al., Nature, 621, 734 (2023)). However, they did not account for photon shot noise or technical noise. Therefore, we believe that our approach is vital for researchers studying optical clocks and cold atoms.

Author Response File: Author Response.pdf

Reviewer 2 Report

In this work,  a general method to extrapolate the number of atoms  used during an atomic clock operation is proposed.
Analyzing  optical clock  instabilities at different configurations, like different trapped atoms and /or different optical probing properties, the authors were able to reconstruct the number of atoms trapped in the optical lattice.
The method   works neglecting the Dick effects, which is the instability induced by the clock laser and considering a constant frequency sensitivity during the different configurations of the clock analyzed.
The authors propose only a numerical investigation but I believe that an experimental comparison can be easily provided.

I found the article well written and  very interesting and I think that  it deserves to be considered for a publication.
I only have a big concern regarding a recently accepted article from the same author (Hong Chang).
I saw that a similar work recently accepted to PRA written by Hong Chang (https://journals.aps.org/pra/accepted/7307aN41Y7c15d2c826b6eb4f2df338e6131f9af2) .
Reading the abstract I see a lot of similarities with this work.   
Unfortunately I am not able to read the pdf to check if there are substantial differences.
Could the authors clarify the differences ?

Concerning the manuscript  I believe that there is a lot of room for improvement for this work.

My main concern is on the beginning of  the  Section 3 ( Result and Discussion).
I strongly suggest to the authors  to split Sec.3 adding a dedicated section for the numerical simulation.
In fact, I believe that the main weak spot of this manuscript is the lack of details regarding the numerical simulation adopted.

I also  provide detailed list of possible improvements for the manuscript:

row 62: In the  previous authors  work (Xiaotong et al, Appl. Phys. Lett. 120, 151104 (2022) https://doi.org/10.1063/5.0085166  )  a different formula for the detection noise was provided.
Could the authors explain why in this work the 1/SNR ( Signal to Noise Ratio) of the detection noise is delta/N_0 while in their previous work was only \delta ?

from row 86 to row106 : I strongly belive that this part needs to be expanded, rewritten more precisely and promoted to a dedicated section for the numerical simulation.
Moreover, it is not clear at all how  the clock is simulated.
A lot of details are not reported in particular what kind of simulation the authors are running.
For me it is not clear at all if the author simulates a time sequence of the clock frequency and how they manage to generate it.

row 88: "In step 1, all the required parameters are initialized and the values \sigma_a ~ \sigma_c are calculated.”
I have no idea what the  “required parameters” are  and the meaning of the symbol “~”.

row 112,113,114: I would promote this comment to the main text of the manuscript

row 120 - 122: it is not clear to me how the authors manage these numerical investigations.

Figure 3 : Would the authors consider a log-log plot ? ( I let the authors to choose the best way to visualize the data)
Moreover, this plot would also be more understandable if a simulated section was provided where the N_OS is properly introduced.

row 144: This statement is not obvious to me.
Why  is the triangular excitation not affected by the excitation fractions fluctuation ?
Could the author be more precise  and expand the sentence a bit?

row 148; How can you achieve a triangular excitation on the atomic sample?
I’m no longer understanding the line.

row 149: function --> works

row 176: Could the authors explain the difference between the N_0 fluctuation and the detection noise ?
What is their physical origin ?
In both cases their SNR scale as deltaN /N  (  \sigma_f  / N)

row200 : A "wide" number of atoms is not a quantitave statement.

Conclusion section: I would appreciate some comments regarding  the role of the S_0.
In this work, the author uses a constant S_0 at different values of \alpha and \beta.
What Do the author expect considering a real-case scenario where their 87Sr clock is running ?
Can S_0 be a function of   \beta  (density broadening)  and how much would be its variation?  

Citation: Typo in the name of [1] Bothwell

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Dear Authors,

Thank you very much for your reply. Indeed, obtaining the exact number of atoms in optical clocks could be challenging. If it is not possible to verification of the numerical data with real data, at minimum it would be worth to describe experimental methods for control a stable atom number by longer time period. Otherwise your numerical model is hard to implement in the real experiment conditions.

Please read the article once again and correct minor typos.

Author Response

Please see the attached file.

Author Response File: Author Response.pdf

Reviewer 2 Report

The authors have provided a satisfactory reply to all my concerns.

They also expand the manuscript with a dedicated section of the simulation which improves the overall readability of the  work.

I have some final comments:

A general comment: The authors often write the words: "measurements time". I would prefer the word "simulated time". It  is not the case of "measurement uncertainty" where the uncertainty is estimated from the simulation result.

row91: How S_0 is computed ?

row 91: Is T_0 kept fixed during all the simulations ? If yes, how is its value ?

row91: Is T_p equal to T_0 in such a way no Dick effect is introduced ?

row 121: I do not believe that eq.1 becomes invalid if S_0 is a function of N_0.

I think that Eq.1 is alway valid if no other noise source is considered ( ex: Dick effect).

I will simply expect that density broadening will reduce  S_0   inducing a  higher instability  for the clock.

Figure 4: After the updated figure version  provided by the authors  I  personally prefer the log-log.

It is evident that relative uncertainty higher than 20% the clock seems not well-locked to the simulated atomic resonance.

Moreover, the inset plot is no longer needed and considerations at N<40 are almost pointless.

Lastly, I do not understand why the updated figure is ambiguous for a large number of atoms.

row 224: measurement time --> simulated time

No comments provided

Author Response

Please see the attached file.

Author Response File: Author Response.pdf