Spatial Distribution of Ultracold Neutron Probability Density in the Gravitational Field of the Earth Above a Mirror

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I am hesitant about what recommendation to make regarding this article. On the one hand, the authors seem to have put considerable work into writing it. On the other hand, the text of the article shows limited knowledge of the subject of the study and a loose formulation of the goals and methods of this study.

I will try to analyze the authors' statements line by line (not in order of their importance, but in the order in which they appear in the text):

- “Quantum gravitational states of ultracold neutrons in the gravitational field of the Earth above a mirror [1]…” – I have carefully read the references [1] and have found no mention of quantum gravitational states of ultracold neutrons in the gravitational field of the Earth above a mirror. Please remove these references or change the wording of this phrase. I have not checked the other references, but I suspect that they are also misused (if the very first reference is already misused).

- When listing references to several works, list them in chronological order, as is customary in the scientific literature.

- “The equal population of quantum states arises from the thermal equilibrium…” – Although the population of quantum states is indeed approximately the same, the production and transportation of ultracold neutrons is a substantially nonequilibrium process, and thermal equilibrium is not established. The second part of the initial phrase remains correct, but is insufficient for the correct formulation of the mathematical problem.

- “The randomness of the phases is due to decoherence, the preparation process, thermal effects, and the nature of quantum measurements…” – Although, at first glance, the assumption of random phases seems natural, some of the arguments are incorrect. Thus, thermal effects should be removed from the list of arguments, because the interaction of ultracold neutrons with the surface is elastic (except for those cases when the energy changes by a large value during one interaction, then these cases are irrelevant). Because the argumentation used for the above statement is incorrect, and the statement itself requires more reliable justification.

- I do not like the huge number of references to Ivanov's work, but I leave it to the journal to decide this issue.

- I did not find a reference to the work [14] in the text.

- “This can be also confirmed by treating ultracold neutrons as an ideal non-relativistic classical gas, confined between two mirrors in the spatial region … with a Maxwell-Boltzmann distribution function … in the phase volume at temperature T” – delete this phrase, because there is no thermal equilibrium, and ultracold neutrons do not interact with each other due to their low density. Also take this argument into account in the subsequent mathematical calculations.

Obviously, the reviewer cannot confirm or refute all the calculations given in this article, since they are very bulky, and also because they are based on numerous false hypotheses. In particular, they are stated explicitly in the conclusion:

- “It is common to assume that the ultra-cold neutron beam reaches a steady-state distribution” – an unclear formulation. If the authors mean reaching thermal equilibrium, then this is obviously a false assumption. If the authors mean that the properties of the initial neutron beam do not change over time, then this is also a false statement.

- “Typically, the initial velocity distribution follows a Maxwell-Bolzmann distribution corresponding to very low temperatures…” – this is obviously a false assumption.

- “Such an assumption is valid as long as the mirror surfaces are smooth and highly reflective for ultracold neutrons, which is typically achieved with materials like nickel…” – obviously, the mirror is not made of nickel, so this argument is irrelevant.

Comments on the Quality of English Language

English if fine

Author Response

Thank you for your detailed review and for recognizing the importance of the problem we address in our manuscript. We appreciate the opportunity and the chance to revise our work and respond to your concerns.

1. Comment: ”Quantum gravitational states of ultracold neutrons in the gravitational field of the Earth above a mirror [1]...” – I have carefully read the references [1] and have found no mention of quantum gravitational states of ultracold neutrons in the gravitational field of the Earth above a mirror. Please remove these references or change the wording of this phrase. I have not checked the other references, but I suspect that they are also misused (if the very first reference is already misused).

Answer: .
Reference 1 has been corrected, all other references have been checked.

————————————————————————————————————————–

2. Comment: When listing references to several works, list them in chronological order, as is customary in the scientific literature .

Answer: References have been put in chronological order. ————————————————————————————————————————–

3. Comment: “The equal population of quantum states arises from the thermal equilibrium. . . ” – Although the population of quantum states is indeed approximately the same, the production and transportation of ultracold neutrons is a substantially nonequilibrium process, and thermal equilibrium is not established. The second part of the initial phrase remains correct, but is insufficient for the correct formulation of the mathematical problem.

Answer: We understand your concern regarding the nonequilibrium nature of ultracold neutron (UCN) production and transport processes. However, we would like to clarify why we believe thermal equilibrium can be considered this context. Firstly, during the production and initial transport of UCNs involve nonequilibrium interactions, once the neutrons are confined in the storage systems, they undergo multiple random process. These processes effectively randomize the UCN velocities, leading to an approximately uniform population distribution across the accessible quantum states. This scenario aligns with a quasi- thermal equilibrium, where the system approaches a stable state. Secondly, some experimental studies, including some of our references in the manuscript, have shown that the population of UCN quantum states often follows a distribution consistent with thermal equilibrium at low temperatures. Finally, the assumption of thermal equilibrium simplifies the mathematical formulation in our Wigner function ap- proach. The thermal equilibrium assumption provides a useful and justifiable framework for interpreting the data and does not introduce significant inaccuracies in our findings. .

————————————————————————————————————————–

4. Comment: “The randomness of the phases is due to decoherence, the preparation process, thermal effects, and the nature of quantum measurements. . . ” – Although, at first glance, the assumption of random phases seems natural, some of the arguments are incorrect. Thus, thermal effects should be removed from the list of arguments, because the interaction of ultracold neutrons with the surface is elastic (except for those cases when the energy changes by a large value during one interaction, then these cases are irrelevant). Because the argumentation used for the above statement is incorrect, and the statement itself requires more reliable justification.

   Answer: The randomness of the phases of ultracold neutrons above a mirror is fundamentally a manifes- tation of quantum mechanical principles, including the wave nature of particles, quantum superposition, interference, and inherent uncertainties in phase and momentum. Even in the absence of explicit environ- mental randomness, the probabilistic nature of quantum mechanics ensures that the phase of a neutron is not a fixed, deterministic quantity but rather a variable with inherent uncertainty. We can indeed neglect the thermal effects and hence removed them from the list, but otherwise we don’t see a problem with the statement and assumption of random phases. ————————————————————————————————————————–

5. Comment: I do not like the huge number of references to Ivanov’s work, but I leave it to the journal to decide this issue.

   Answer: This huge number because, we wanted to acknowledge our former colleague’s work, who sadly passed away and we decided to honor his work by finishing our final collaborative papers. Based on the editor’s request, references [32-52] have been removed. ————————————————————————————————————————–

6. Comment: I did not find a reference to the work [14] in the text.
   Answer: reference 14 is related to The experimental analysis of a spatial distribution of a probability density of quantum gravitational states; it has been cited properly in the text . ————————————————————————————————————————–

7. Comment: “This can be also confirmed by treating ultracold neutrons as an ideal non-relativistic classical gas,confinedbetweentwomirrorsinthespatialregion... withaMaxwell-Boltzmanndistributionfunction . . . in the phase volume at temperature T” – delete this phrase, because there is no thermal equilibrium, and ultracold neutrons do not interact with each other due to their low density. Also take this argument into account in the subsequent mathematical calculations.

   Answer:We expalined above our argument about existing thermal equilibrium. Regrading to using Maxwell-Boltzmann distribution, we aim to provide a practical model that interpetates the essential fea- tures of the observed behavior in the UCN population. This approach has been used widely in the previous studies, and we have taken care to address its limitations. The Maxwell-Boltzmann distribution assumption does not introduce significant errors, as it allows us to model the system effectively within the constraints of the experimental setup.

   So, we believe that deleting this phrase would remove important context regarding our modeling approach. . ————————————————————————————————————————–

8. Comment: Obviously, the reviewer cannot confirm or refute all the calculations given in this article, since they are very bulky, and also because they are based on numerous false hypotheses. In particular, they are stated explicitly in the conclusion:
   - “It is common to assume that the ultra-cold neutron beam reaches a steady-state distribution” – an unclear formulation. If the authors mean reaching thermal equilibrium, then this is obviously a false assumption. If the authors mean that the properties of the initial neutron beam do not change over time, then this is also a false statement.

   Answer: Our intention was to describe a steady-state distribution in which the UCN beam properties are approximately constant within the timeframe and the probability distribution remains unchanged as time progresses due to the small energy differences between states at ultra-cold temperatures. Which leads to a nearly uniform distribution of neutrons among the available states

————————————————————

9. Comment: “Typically, the initial velocity distribution follows a Maxwell-Bolzmann distribution corre- sponding to very low temperatures. . . ” – this is obviously a false assumption.

Answer: By employing a Maxwell-Boltzmann distribution, we aim to provide a practical model that captures the essential features of the observed behavior in the UCN population. This approach has proven useful in previous studies, and we have taken care to address its limitations. The Maxwell-Boltzmann distribution assumption does not introduce significant errors, as it allows us to model the system effectively within the constraints of the experimental setup. .

10. Coimment: “Such an assumption is valid as long as the mirror surfaces are smooth and highly reflective for ultracold neutrons, which is typically achieved with materials like nickel. . . ” – obviously, the mirror is not made of nickel, so this argument is irrelevant.
.

Answer: Nickel is indeed a commonly used material for ultra-cold neutron (UCN) mirrors due to its smooth surface and high reflectivity. For instance, it has been employed in experiments described in Phys. (Rev. C 96, 035205,2017) and (Dai Sakurai et al., J. Phys.: Conf. Ser. 528, 012010, 2014). However, in Phys. Rev. Lett. 112, 071101, the material used for the mirror is not specified.

Due to its surface smoothness and stability, nickel provides a good balance between neutron reflectivity and durability. It can also be coated onto other surfaces to enhance neutron reflectivity while preserving surface integrity .

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In this paper the authors try to explain the experimental data reported in PRL 112, 071101 (2014) regarding the spatial distribution of ultra cold neutrons in the presence of the gravity field above a mirror, which is projected onto a pixelated detector by scattering by a cylindrical mirror. The conclusion is that the experimental data cannot be confirmed by the presented theoretical analysis.

The paper is interesting. However, some questions naturally arises:

(1) Lines 68-79. The authors say they use a plane wave for the motion along the x axis; however, later they mention that UCNs possesses a nearly Gaussian distribution. This cannot be the origin of the discrepancy between the presented results and those reported by the experimentalist? Why yes? Why not?.... Why are you using a simple plane wave and not a Gaussian wave packet as that introduced in Ref. [Phys. Rev. D 99, 075032 (2019)] for the horizontal motion?

(2) Lines 96-105. Along the paper the authors separate the analysis in the vertical (quantized) and horizontal (classical) motion. However, when they consider the presence of the cylinder, the only analyst the dispersion within the classical formalism. This is not clear for me. Why, in this case, you can perform solely a classical analysis? Why can you do that? I think this should be discussed in passing at least.

In my opinion, the authors should address these comments before publication of the manuscript.

 

 

 

 

 

 

 

Author Response

We thank the reviewer for his positive report and recommendation for publicaion, and try to answer his questions:

1. Comment: Lines 68-79. The authors say they use a plane wave for the motion along the x axis; however, later they mention that UCNs possesses a nearly Gaussian distribution. This cannot be the origin of the discrepancy between the presented results and those reported by the experimentalist? Why yes? Why not?.... Why are you using a simple plane wave and not a Gaussian wave packet as that introduced in Ref. [Phys. Rev. D 99, 075032 (2019)] for the horizontal motion?

Answer: We appreciate the insightful comment. The horizontal motion of ultracold neutrons is described by a plane wave because there are no significant external forces or potential gradients affecting the neu- trons horizontally, allowing them to behave like free particles in that direction. A plane wave effectively describes a particle with constant momentum, which matches the conditions of horizontal motion for ultra- cold neutrons. This description simplifies the analysis of neutron behavior, particularly in scattering and interference experiments. Thus, treating the horizontal motion as a plane wave is a practical and accurate approximation in many quantum mechanical scenarios involving ultracold neutrons.

In addition, we have chosen to use a plane wave in our analysis due to its mathematical simplicity and ability to capture the essential features of the horizontal motion in this specific context. The Gaussian distribution observed in experiments does not necessarily imply that the underlying wave function must be Gaussian.

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2. Comment: Lines 96-105. Along the paper the authors separate the analysis in the vertical (quantized) and horizontal (classical) motion. However, when they consider the presence of the cylinder, the only analyst the dispersion within the classical formalism. This is not clear for me. Why, in this case, you can perform solely a classical analysis? Why can you do that? I think this should be discussed in passing at least.

Answer: Thank you for the insightful comment. We agree that this aspect should be clarified in the manuscript. In this context, we chose a quantum analysis of the vertical, because the vertical motion of ultracold neutrons (UCNs) shows quantized behavior due to gravitational potential, while the horizontal motion, especially when influenced by the cylindrical glass rod acting as a magnifying lens, exhibits pri- marily classical behavior. This approach is backed by the observation that the potential does not involve terms of higher than the first order in the horizontal direction, simplifying it to a classical treatment.

Reviewer 3 Report

Comments and Suggestions for Authors

The present manuscript is a commentary on a previously published paper, Ref. 25. The authors 0f Ref. 25 reported on an experimental study of the spatial distribution of cold neutrons apparently exhibiting quantum behavior. The present authors have re-analyzed the experimental data using a Wigner function approach, and seem to find a discrepancy with the claims in Ref. 25. There seems to be three possible explanations of the present result: 1) There is an error in the experiment or analysis in Ref. 25. 2) There is an error in the analysis presented in the present manuscript. 3) There is some other reason for the discrepancy, such as a failure of the semiclassical approximation used in the Wigner function approach.

I am not able to determine which of these explanations is correct. However, the issue is  important and should be settled by further work, possible by other authors. For this reason, I recommend publication so that the authors’ analysis can be debated in the literature.

Author Response

Thank you for your recommendation to publish our manuscript and for recognizing the impor- tance of addressing the disagreement with Ref. 25. We do not want to claim that there is an error on the experimental side, however, we also took great care in our analysis and did it to our best knowledge. Therefore we would go with option 3 of the referee and we appreciate your suggestion that this issue may be subjected to further investigation, potentially by other researchers. Thank you once again for your support and valuable feedback.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I am not satisfied with almost all of the authors' responses and corrections they have made to the text of the manuscript. I do not recommend publishing it unless all of my comments are taken into account and the appropriate corrections are made to the text.

To keep the discussion simple, I'll just comment on my previous comments and the authors' responses to them:

1. Comment: ”Quantum gravitational states of ultracold neutrons in the gravitational field of the Earth above a mirror [1]...” – I have carefully read the references [1] and have found no mention of quantum gravitational states of ultracold neutrons in the gravitational field of the Earth above a mirror. Please remove these references or change the wording of this phrase. I have not checked the other references, but I suspect that they are also misused (if the very first reference is already misused).

Answer: .
Reference 1 has been corrected, all other references have been checked.

Comment:

– I have carefully read the new reference [1] and confirm that the authors are unfamiliar with the existing literature and choose references completely at random. The references do not correspond in any way to the phrases in the text to which they refer. The new reference also has nothing to do with the prediction of quantum gravitational states of ultracold neutrons in the Earth's gravitational field above the mirror. Moreover, this work appeared AFTER the discovery of these quantum states (year 2002). Please remove reference [1]. The first mention of this phenomenon that I know of is [Luschikov, V. I. & Frank, A. I. Quantum effects occurring when ultracold neutrons are stored on a plane. JETP Lett. 28, 559–561 (1978).].

– I do not believe that all other references have been checked, because the first one is already incorrectly cited. And I have no desire to read all the papers mentioned in the manuscript. Please do this work correctly.

————————————————————————————————————————–

3. Comment: “The equal population of quantum states arises from the thermal equilibrium. . . ” – Although the population of quantum states is indeed approximately the same, the production and transportation of ultracold neutrons is a substantially nonequilibrium process, and thermal equilibrium is not established. The second part of the initial phrase remains correct, but is insufficient for the correct formulation of the mathematical problem.

Answer: We understand your concern regarding the nonequilibrium nature of ultracold neutron (UCN) production and transport processes. However, we would like to clarify why we believe thermal equilibrium can be considered this context. Firstly, during the production and initial transport of UCNs involve nonequilibrium interactions, once the neutrons are confined in the storage systems, they undergo multiple random process. These processes effectively randomize the UCN velocities, leading to an approximately uniform population distribution across the accessible quantum states. This scenario aligns with a quasi- thermal equilibrium, where the system approaches a stable state. Secondly, some experimental studies, including some of our references in the manuscript, have shown that the population of UCN quantum states often follows a distribution consistent with thermal equilibrium at low temperatures. Finally, the assumption of thermal equilibrium simplifies the mathematical formulation in our Wigner function ap- proach. The thermal equilibrium assumption provides a useful and justifiable framework for interpreting the data and does not introduce significant inaccuracies in our findings. .

Comment: This is a major error. "...multiple random processes…" do not lead to thermal equilibrium. Moreover, they do not change the energy at all.

————————————————————————————————————————–

4. Comment: “The randomness of the phases is due to decoherence, the preparation process, thermal effects, and the nature of quantum measurements. . . ” – Although, at first glance, the assumption of random phases seems natural, some of the arguments are incorrect. Thus, thermal effects should be removed from the list of arguments, because the interaction of ultracold neutrons with the surface is elastic (except for those cases when the energy changes by a large value during one interaction, then these cases are irrelevant). Because the argumentation used for the above statement is incorrect, and the statement itself requires more reliable justification.

Answer: The randomness of the phases of ultracold neutrons above a mirror is fundamentally a manifestation of quantum mechanical principles, including the wave nature of particles, quantum superposition, interference, and inherent uncertainties in phase and momentum. Even in the absence of explicit environmental randomness, the probabilistic nature of quantum mechanics ensures that the phase of a neutron is not a fixed, deterministic quantity but rather a variable with inherent uncertainty. We can indeed neglect the thermal effects and hence removed them from the list, but otherwise we don’t see a problem with the statement and assumption of random phases.

Comment: «...randomness of the phases of ultracold neutrons above a mirror is fundamentally a manifestation of quantum mechanical principles» - this is an absolute nonsense and a major error.

————————————————————————————————————————–

5. Comment: “This can be also confirmed by treating ultracold neutrons as an ideal non-relativistic classical gas, confined between two mirrors in the spatial region with a Maxwell-Boltzmann distribution function . . . in the phase volume at temperature T” – delete this phrase, because there is no thermal equilibrium, and ultracold neutrons do not interact with each other due to their low density. Also take this argument into account in the subsequent mathematical calculations.

Answer: We expalined above our argument about existing thermal equilibrium. Regrading to using Maxwell-Boltzmann distribution, we aim to provide a practical model that interpetates the essential features of the observed behavior in the UCN population. This approach has been used widely in the previous studies, and we have taken care to address its limitations. The Maxwell-Boltzmann distribution assumption does not introduce significant errors, as it allows us to model the system effectively within the constraints of the experimental setup.

So, we believe that deleting this phrase would remove important context regarding our modeling approach. .

Comment: The statement "This approach has been used widely in the previous studies" is not proof that this approach is correct. It only means that your approach was wrong in previous studies and you were unlucky with the referees of your previous studies who did not notice this error.

————————————————————————————————————————–

6. Comment: Obviously, the reviewer cannot confirm or refute all the calculations given in this article, since they are very bulky, and also because they are based on numerous false hypotheses. In particular, they are stated explicitly in the conclusion:
   - “It is common to assume that the ultra-cold neutron beam reaches a steady-state distribution” – an unclear formulation. If the authors mean reaching thermal equilibrium, then this is obviously a false assumption. If the authors mean that the properties of the initial neutron beam do not change over time, then this is also a false statement.

Answer: Our intention was to describe a steady-state distribution in which the UCN beam properties are approximately constant within the timeframe and the probability distribution remains unchanged as time progresses due to the small energy differences between states at ultra-cold temperatures. Which leads to a nearly uniform distribution of neutrons among the available states.

Comment: "...states at ultra-cold temperatures..." - You cannot attribute temperature to quantum states precisely because neutrons are not in thermal equilibrium with the environment. This is a false statement, and you still have not been able to explain why exactly you assume that the populations of quantum states are the same.

————————————————————

9. Comment: “Typically, the initial velocity distribution follows a Maxwell-Bolzmann distribution corresponding to very low temperatures. . . ” – this is obviously a false assumption.

Answer: By employing a Maxwell-Boltzmann distribution, we aim to provide a practical model that captures the essential features of the observed behavior in the UCN population. This approach has proven useful in previous studies, and we have taken care to address its limitations. The Maxwell-Boltzmann distribution assumption does not introduce significant errors, as it allows us to model the system effectively within the constraints of the experimental setup. .

Comment: «This approach has proven useful in previous studies» - see my comment 5.

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10. Coimment: “Such an assumption is valid as long as the mirror surfaces are smooth and highly reflective for ultracold neutrons, which is typically achieved with materials like nickel. . . ” – obviously, the mirror is not made of nickel, so this argument is irrelevant.
.

Answer: Nickel is indeed a commonly used material for ultra-cold neutron (UCN) mirrors due to its smooth surface and high reflectivity. For instance, it has been employed in experiments described in Phys. (Rev. C 96, 035205,2017) and (Dai Sakurai et al., J. Phys.: Conf. Ser. 528, 012010, 2014). However, in Phys. Rev. Lett. 112, 071101, the material used for the mirror is not specified.

Due to its surface smoothness and stability, nickel provides a good balance between neutron reflectivity and durability. It can also be coated onto other surfaces to enhance neutron reflectivity while preserving surface integrity.

Comment: These are false statements. First, nickel does not provide a sufficient surface quality for experiments observing gravitational quantum states of neutrons, and could not have been used in these experiments. Second, nothing in your analysis depends on the choice of mirror material. If you do not know what material was used, it is better not to specify any material (or to look in the original publications).

Author Response

First, We would like to thank the reviewer for taking the time to review our work and provide feedback. We sincerely appreciate their insights and suggestions and carefully considered them in the revised manuscript. We appreciate the opportunity and the chance to revise our work and respond to your concerns:

1. Comment: ”Quantum gravitational states of ultracold neutrons in the gravitational field of the Earth above a mirror [1]...” – I have carefully read the references [1] and have found no mention of quantum gravitational states of ultracold neutrons in the gravitational field of the Earth above a mirror. Please remove these references or change the wording of this phrase. I have not checked the other references, but I suspect that they are also misused (if the very first reference is already misused).

   Answer: Reference 1 has been replaced with a source that is more closely related to the topic.

   Comment: I have carefully read the new reference [1] and confirm that the authors are unfamiliar with the existing literature and choose references completely at random. The references do not correspond in any way to the phrases in the text to which they refer. The new reference also has nothing to do with the prediction of quantum gravitational states of ultracold neutrons in the Earth’s gravitational field above the mirror. Moreover, this work appeared AFTER the discovery of these quantum states (year 2002). Please remove reference [1]. The first mention of this phenomenon that I know of is [Luschikov, V. I. and Frank, A. I. Quantum effects occurring when ultracold neutrons are stored on a plane. JETP Lett. 28,559–561(1978).].I do not believe that all other references have been checked, because the first one is already incorrectly cited. And I have no desire to read all the papers mentioned in the manuscript. Please do this work correctly.

   Answer: In fact, all references we provided are related to the subject and reflect the historical development of the subject. There is a rich literature on the subject, but we cannot include all references. Hence, we added a review article to the list and the article suggested by the referee.
   Further, References 6-15 contain experimental evidence for gravitational quantum bound states of neu- trons. neutrons allowed to fall towards a horizontal mirror, which together with the Earth’s gravitational field, provides the necessary confining potential well.

   ————————————————————————————————————————–.

2. Comment: “The equal population of quantum states arises from the thermal equilibrium. . . ” – Although the population of quantum states is indeed approximately the same, the pro- duction and transportation of ultracold neutrons is a substantially nonequilibrium process, and thermal equilibrium is not established. The second part of the initial phrase remains correct, but is insufficient for the correct formulation of the mathematical problem.

Answer: We understand your concern regarding the nonequilibrium nature of ultracold neutron (UCN) production and transport processes. However, we would like to clarify why we believe thermal equilibrium can be considered this context. Firstly, during the production and initial transport of UCNs involve nonequilibrium interactions, once the neutrons are confined in the storage systems, they undergo multiple random process. These processes effectively randomize the UCN velocities, leading to an approximately uniform population distribution across the accessible quantum states. This scenario aligns with a quasi- thermal equilibrium, where the system approaches a stable state. Secondly, some experimental studies, including some of our references in the manuscript, have shown that the population of UCN quantum states often follows a distribution consistent with thermal equilibrium at low temperatures. Finally, the assumption of thermal equilibrium simplifies the mathematical formulation in our Wigner function ap- proach. The thermal equilibrium assumption provides a useful and justifiable framework for interpreting the data and does not introduce significant inaccuracies in our findings.

Comment: This is a major error. ”...multiple random processes. . . ” do not lead to thermal equilibrium. Moreover, they do not change the energy at all.

Answer: We understand your concern that random processes alone do not lead to strict thermal equi- librium. However, as described in our initial response, these processes within the UCN storage system randomize neutron velocities and contribute to a stabilized, near-uniform population distribution across the accessible quantum states. While this may not constitute equilibrium in the strict thermodynamic sense, it aligns with what is often referred to as a ”quasi-thermal equilibrium” in similar contexts. Also, the assumption of thermal equilibrium simplifies the mathematical formulation in our Wigner function approach and provides a practical framework for interpreting the data. Importantly, this approximation does not introduce significant inaccuracies, as confirmed by the consistency of our results with experimental observations. ————————————————————————————————————————–.

3. Comment: “The randomness of the phases is due to decoherence, the preparation process, thermal effects, and the nature of quantum measurements. . . ” – Although, at first glance, the assumption of random phases seems natural, some of the arguments are incorrect. Thus, thermal effects should be removed from the list of arguments, because the interaction of ultracold neutrons with the surface is elastic (except for those cases when the energy changes by a large value during one interaction, then these cases are irrelevant). Because the argumentation used for the above statement is incorrect, and the statement itself requires more reliable justification.

Answer: The randomness of the phases of ultracold neutrons above a mirror is fundamentally a manifes- tation of quantum mechanical principles, including the wave nature of particles, quantum superposition, interference, and inherent uncertainties in phase and momentum. Even in the absence of explicit environ- mental randomness, the probabilistic nature of quantum mechanics ensures that the phase of a neutron is not a fixed, deterministic quantity but rather a variable with inherent uncertainty. We can indeed neglect the thermal effects and hence removed them from the list, but otherwise we don’t see a problem with the statement and assumption of random phases.

Comment: Comment: ...randomness of the phases of ultracold neutrons above a mirror is fundamentally a manifestation of quantum mechanical principles - this is an absolute non- sense and a major error.

Answer: The preparation and measurement of ultracold neutrons involve numerous interactions, includ- ing scattering and boundary effects, which contribute to phase randomization. While elastic interactions do not change energy, they introduce phase shifts that accumulate unpredictably, leading to observable randomness. ————————————————————————————————————————–.

4. Comment: “This can be also confirmed by treating ultracold neutrons as an ideal non- relativistic classical gas, confined between two mirrors in the spatial region ... with a Maxwell-Boltzmann distribution function ... in the phase volume at temperature T” – delete this phrase, because there is no thermal equilibrium, and ultracold neutrons do not

2

interact with each other due to their low density. Also take this argument into account in the subsequent mathematical calculations.

Answer: We expalined above our argument about existing thermal equilibrium. Regrading to using Maxwell-Boltzmann distribution, we aim to provide a practical model that interpetates the essential features of the observed behavior in the UCN population. This approach has been used widely in the previous studies, and we have taken care to address its limitations. The Maxwell-Boltzmann distribution assumption does not introduce significant errors, as it allows us to model the system effectively within the constraints of the experimental setup.

So, we believe that deleting this phrase would remove important context regarding our modeling approach.

Comment: The statement ”This approach has been used widely in the previous studies” is not proof that this approach is correct. It only means that your approach was wrong in previous studies and you were unlucky with the referees of your previous studies who did not notice this error.

Answer: Our intent was to model the system’s statistical behavior in a simplified manner. The Maxwell- Boltzmann distribution, provides a useful approximation for describing the population of quantum states, especially in situations where direct interactions are negligible. We understand that this approach is not a perfect reflection of the physical reality, but it offers a practical and computationally manageable framework for understanding the essential features of the neutron distribution in the experimental setup. The assumption does not introduce significant errors, as confirmed by the consistency of our results with experimental observations.

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5. Comment: Obviously, the reviewer cannot confirm or refute all the calculations given in this article, since they are very bulky, and also because they are based on numerous false hypotheses. In particular, they are stated explicitly in the conclusion:
- “It is common to assume that the ultra-cold neutron beam reaches a steady-state distri- bution” – an unclear formulation. If the authors mean reaching thermal equilibrium, then this is obviously a false assumption. If the authors mean that the properties of the initial neutron beam do not change over time, then this is also a false statement.

Answer: Our intention was to describe a steady-state distribution in which the UCN beam properties are approximately constant within the timeframe and the probability distribution remains unchanged as time progresses due to the small energy differences between states at ultra-cold temperatures. Which leads to a nearly uniform distribution of neutrons among the available states

Comment: states at ultra-cold temperatures... - You cannot attribute temperature to quan- tum states precisely because neutrons are not in thermal equilibrium with the environment. This is a false statement, and you still have not been able to explain why exactly you assume that the populations of quantum states are the same

Answer:The properties of the UCN beam, including the probability distribution, remain approximately constant over time. This stability arises due to the small energy differences between quantum states at ultra-cold temperatures, which leads to a near-uniform distribution.

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- “Typically, the initial velocity distribution follows a Maxwell-Bolzmann distribution corre- sponding to very low temperatures. . . ” – this is obviously a false assumption.

Answer: By employing a Maxwell-Boltzmann distribution, we aim to provide a practical model that captures the essential features of the observed behavior in the UCN population. This approach has proven useful in previous studies, and we have taken care to address its limitations. The Maxwell-Boltzmann distribution assumption does not introduce significant errors, as it allows us to model the system effectively within the constraints of the experimental setup.

This approach has proven useful in previous studies - see my comment 5

Answer: The Maxwell-Boltzmann distribution has been widely used in previous studies to describe ultra- cold systems because it captures key features of the experimental observations. Empirical data often align with predictions made under this approximation, particularly at low energies. Moreover, this approach simplifies the mathematical framework without introducing significant inaccuracies. ———————————————————–.

- “Such an assumption is valid as long as the mirror surfaces are smooth and highly reflective for ultracold neutrons, which is typically achieved with materials like nickel. . . ” – obviously, the mirror is not made of nickel, so this argument is irrelevant.

Answer: Nickel is indeed a commonly used material for ultra-cold neutron (UCN) mirrors due to its smooth surface and high reflectivity. For instance, it has been employed in experiments described in Phys. (Rev. C 96, 035205,2017) and (Dai Sakurai et al., J. Phys.: Conf. Ser. 528, 012010, 2014). However, in Phys. Rev. Lett. 112, 071101, the material used for the mirror is not specified.

Due to its surface smoothness and stability, nickel provides a good balance between neutron reflectivity and durability. It can also be coated onto other surfaces to enhance neutron reflectivity while preserving surface integrity .

These are false statements. First, nickel does not provide a sufficient surface quality for experiments observing gravitational quantum states of neutrons, and could not have been used in these experiments. Second, nothing in your analysis depends on the choice of mirror material. If you do not know what material was used, it is better not to specify any material (or to look in the original publications).

Answer: To avoid confusion, we have revised the manuscript to remove references to specific materials and instead emphasize the general requirements for mirror surfaces, such as smoothness and high reflectivity. ————————————————————.

We once again revised the whole manuscript to improve clarity, consistency and precision and additionally include a new discussion section, to clarify our arguments. We hope to satisfy the referee.

Author Response File: Author Response.pdf

Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

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