Larmor Time for Trapezoidal Barrier

Round 1

Reviewer 1 Report (New Reviewer)

Comments and Suggestions for Authors

Please see the attached pdf file

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report (Previous Reviewer 1)

Comments and Suggestions for Authors

Due to the author's excellent response to my question, I suggest publishing this paper.

Author Response

We sincerely appreciate your valuable assistance and dedication to this manuscript.

Reviewer 3 Report (New Reviewer)

Comments and Suggestions for Authors

This manuscript presents a detailed theoretical investigation into the Larmor tunneling time for three types of trapezoidal potential barriers. The authors have systematically calculated the transmission and reflection Larmor times under two distinct configurations: the traditional Baz-Rybachenko-Büttiker setup and a new Xiao-Zheng-Liu configuration. 

While the work is thorough, and the methodology appears sound, several fundamental points regarding the theoretical framework and the interpretation of the results require clarification and significant revision before the paper can be considered for publication. The specific points are detailed below. Major Points for Revision

1.Motivation and Context in the Introduction: The introduction provides a general background on tunneling time but lacks a sufficiently compelling motivation for this specific study. The authors should expand the introduction by reviewing the relevant literature on Larmor time in more detail to better contextualize their work and the motivation. 

2.Clarification of the Potential Barrier Setup: In the description of the potential barriers and in the scattering solution (Eq. 2), the manuscript introduces a point z = b (with b < 0). However, its role is not entirely clear from the diagrams in Figure 2, where the barriers appear to begin at z = 0. On page 3, line 73, it is briefly mentioned that the magnetic field is extended to the left for reflection calculations. This needs to be explained more explicitly. Please clarify the physical and mathematical necessity of the region z < b and b < z < 0. 

3.The Weak-Field Approximation: The Larmor clock is a perturbative concept, valid in the weak-field limit. This crucial assumption is not explicitly stated or discussed in the manuscript. The authors should indicate where the weak-field approximation is applied in their derivation.

4.Physical Interpretation of Consistency: The paper correctly highlights that the two different configurations (B and X) yield "almost indistinguishable" results for the Larmor times. This is a non-trivial finding. However, the manuscript stops at reporting this consistency without offering a physical interpretation. What is the underlying physical reason for this consistency, considering that the two approaches are derived from changes in different angles? The paper would be significantly strengthened by a discussion of the underlying physical reasons for this agreement.

5.Interpretation of Negative Reflection Times: In Figures 4(b) and 5(b), the reflection Larmor time (τ_R) takes on negative values in certain regimes. While this is a known feature in tunneling time literature, it is a counter-intuitive result that warrants explanation for the reader. 

6.Qualitative Explanation of Dynamic Features: The manuscript presents plots showing peaks and oscillations but could benefit from a deeper qualitative analysis of these phenomena.

(1)Peak Structure: Both transmission (τ_T) and reflection (τ_R) times exhibit distinct peaks at certain wave numbers. What is the qualitative physical origin of these peaks? 

(2)High-Energy Oscillations: The reflection time τ_R shows rapid oscillations at larger wave numbers. While the conclusion touches upon this, a qualitative explanation is needed for this oscillatory behavior. 

In summary, this manuscript addresses a relevant problem in quantum tunneling with a thorough computational approach. However, in its current form, the paper requires major revisions. The issues raised above concerning the theoretical framework, the interpretation of the results, and the clarity of the presentation are significant and must be addressed. 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report (New Reviewer)

Comments and Suggestions for Authors

The authors have answered all my questions and comments and modified the manuscript satisfactorily. The article in my opinion is suitable for publication in Atoms.

Author Response

Thank you for your guidance and help with our manuscript.

Reviewer 3 Report (New Reviewer)

Comments and Suggestions for Authors

The authors have provided a detailed response to the initial review and have revised the manuscript. The added explanations have significantly improved the clarity and logical flow of the paper.

We have one remaining point of discussion concerning the potential barrier setup at . The manuscript now states this is necessary to define a region for the reflection process, which carries the "path information" of the reflected particles. However, we believe this definition warrants further consideration. In the seminal literature by Büttiker [22], the reflection time is discussed without the introduction of such an additional boundary to the left of the barrier. More recent experimental work [18] also successfully investigates particle interaction time within a well-defined barrier region without needing to extend it in this manner. Therefore, we are curious if setting b = 0 for the reflection calculation, what the outcome would be.

In summary, the revised manuscript is clearer and provides a deeper insight into the topic of Larmor tunneling time across various potential barrier configurations. We recommend that the paper be accepted for publication.

Author Response

Thank you for your guidance and help with our manuscript.

According to the work of Falck and Hauge (1988), in order to obtain reliable results for the measurement of tunneling time using a Larmor clock, the reading itself should be performed in a region far from the barrier. Moreover, it is crucial that the region where the weak magnetic field is applied must be sufficiently wide to cover and be larger than the barrier region itself, thereby effectively avoiding interference caused by quantum interference effects [31]. Therefore, in our calculations, we present the reflected Larmor results for particle tunneling under the condition that the magnetic field region is larger than the barrier region. Although the current research focuses on particle tunneling (rather than wave packets) with a monochromatic approximation such that the spectrum function $g(k)\propto\delta(k-k_0)$ (where $k_0$ is the mean wave number of the incoming particle), the results presented in this paper will also provide an important reference for the subsequent in-depth study of wave packet tunneling dynamics. In view of your interest in the reflection results under the condition of b=0, we hereby specially provide some reflection results under b=0 for your reference. In other words, we may view $b=0$ as a limiting case for a more strict wave packet treatment in terms of Larmor time.

Author Response File: Author Response.pdf

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this paper, the authors explore the Larmor time for three types of trapezoidal barriers, and they find consistent results between the traditionally defined Larmor time and a newly defined one. They confirm that the transmission Larmor time for the trapezoidal barriers also satisfies certain properties of mirror-symmetric barriers. They also find that as the barrier height increases, the peak of the Larmor time shifts to the right, and as the barrier width increases, the peak becomes larger in coincidence with the classical expectation that a particle needs more time to cross a longer path of the same height/inclination.

 

These results make sense and sound interesting, and I believe can be of interest to Atoms readers as well. I can therefore recommend this manuscript for publication in Atoms once the authors address the following points:

 

1.There are some grammatical errors, such as the misspelling “tunnling” in line 45 of page 2 should be changed to “tunneling”, and so on. I recommend the authors to correct these grammatical errors to make the paper more clear for readers to read.

2.There are some citation errors. For example, incorrect the reference of the Fig. 1 in 138 of page 6 should be changed to Fig. 2, and so on.. I suggest the authors examine the manuscript carefully.

3.How to derive Eq.(8) in detail. Please the authors give detail derivations processes.

4.The authors should briefly discuss the impact of external environments (including both Markovian and non-Markovian effects) on their results as the paper primarily considers closed systems [PRA 109, 023712 (2024); PRA 99, 032101 (2019); PRA 98, 062106 (2018); PRL 103, 210401 (2009); RMP 88, 021002 (2016); PRA 31, 3761 (1985)].

5.The authors are encouraged to briefly address how this scheme can be implemented in current experimental setups and provide relevant references to support the discussion.

Reviewer 2 Report

Comments and Suggestions for Authors

This paper investigates the Larmor tunneling times for three types of ladder potential barriers from a theoretical perspective, using two different spin-magnetic field configurations. The study extends the Larmor time formalism to more complex barrier geometries. Numerical results compare the transmission and reflection Larmor times for different barrier parameters. The paper seems technically useful. I recommend acceptance after minor revisions to improve clarity, presentation, and depth of explanation of the Discussion, for example, more discussion of the physical picture of the Larmor time trends, and the importance of oscillations or the sharp discontinuities in the reflection times, would improve accessibility.