Saturation Effect Produced by Laser Pulses: Karplus–Schwinger Approach versus Bloch Solution

Round 1

Reviewer 1 Report

 

Comments for author File: Comments.pdf

Author Response

First of all, we would like to express our gratitude to the Reviewer for his/her valuable work. We are confident that corrections based on his/her comments will make our article better.

 

1–  The authors focused in the introduction on the literature review of the saturation effect, however they did not explain the interactive part between the laser and the two-level system. (although they mentioned in paragraph 3 (starting line 9 of the introduction) a general sentence)

One scientific paragraph might be of great benefit to the readers, especially for young PhD students, hence the authors should explain why should we get exact solutions of the Bloch equations?

As a hint, the answer to this is that the Bloch equations are useful in the fundamental mathematical physics like, quantum optics, quantum control as well as many other topics. Another pertinent field where we need to derive the exact solutions for the Bolch equations is the NMR (medical physics). The authors are encouraged to cite few pertinent articles within this paragraph

We insert several sentences into Introduction devoted to this subject highlighted in yellow and corresponding references 4-6 (in new numeration)

 

 2- it is important to explain why the time interaction between the pulse and the two level atom is of great importance:  

We insert several sentences highlighted in yellow into Introduction devoted to this subject and corresponding references 7-8 (in new numeration)

3- Solving the bloch equation is another point that was not covered in the introduction: other pertinent references should be mentioned example: Exact solution of the Bloch equations for the nonresonant exponential model in the presence of dephasing

1. N. Zlatanov, G. S. Vasilev, P. A. Ivanov, and N. V. Vitanov Phys. Rev. A 92, 043404 – Published 6 October 2015

4-

Author might read about the shaped pulses investigated in a two level atom where the exact solution for the Bloch equation was derived. With such pulses, not only the exponential is used, but also other waveforms for pertinent reasons. Many examples are available online, for the recent investigations like :

Example:

Analysis of a q-deformed hyperbolic short laser pulse in a multi-level atomic system N Boutabba, S Grira, H Eleuch Scientific Reports 12 (1), 1-7) 2022. Nature

We insert several sentences highlighted in yellow into Introduction devoted to this subjects and corresponding references 9-11 (in new numeration)

5- What is the unit of tau in all figures? Is it nano seconds?

All variables in the paper are in relative units. It is possible because the probabilities (10)–(12) and Bloch equations (14)–(16) are invariant relative to the following transformation:

Frequency variables ® Frequency variables/ws

Time variables ® Time variables´ws

Here ws is some scaling frequency.

For example if ws=atomic frequency=4.13×10¹⁶ s^-1 then t and all time parameters are in atomic units, i.e. relative time unit=2.4×10^-17 s.

We add corresponding paragraph highlighted in yellow on page 6.

6- In the last line of the article, the authors state that the same conclusions hold for lasers with other enveloppes: what are the other enveloppes? Any example was studied a part from the exponential pulse?

We also studied double exponential and Gaussian envelope. We consider in the paper exponential envelope because in this case it is possible to derive simple analytical expressions for excitation probability.

We add corresponding sentence highlighted in yellow in the end of the Summary.

7- For next submissions and revisions, it is recommended that the authors use numbering of lines

Done

8- The author has great and extensive work in the subject of the study. However, 66% of the articles cited belongs to the author, and as per the standard of the journal, we strongly advice and recommend the reduction of the self-citation.

We removed references 10-13 and 18 (in initial numeration) and added 9 new references on another authors.

 

 

Author Response File: Author Response.pdf

Reviewer 2 Report

 

The manuscript

“Saturation effects….”

By Astapenko & Lisitsa

 

report about the study of ultra-short pulse effects in photo-excitation and the comparison of different methods of calculations, namely the exact Bloch equations, the Karplus-Schwinger approach with various types of approximations.

In view of the rapid evolution of short pulse laser systems, the manuscript is of great interest for the community. The paper is physically very well constructed and should be published in ATOMS.

 

Before publication, several language corrections and clarifications of formulas and units should be inserted:

 

1. Eq. (1) and following: what units are used ?

 

2. Concerning eqs. (14-16):

- R1,2,3 are not defined: write in first phrase after Bloch vector the expression (R1,R2,R3)

- write T1 and T2 are the relaxation times of levels 1 and 2, respectively.

 

3.  Eq. 17: provide the information in what interval R3 can be, 0-1 ? For consistency, write also about R1, R2.

 

4. Last phrase of paragraph 4: why T1->infinity is correct ? Add short explanation

 

5. Paragraph 5: what means relative units? Relative to what ? Same for all figures captions..Provide information how a value of tau, t1, Omega_0 can be transformed to seconds and angular frequency.

 

6. It remains unclear in the figures what GKSA means, is it numerical solution of eqs. (1-4) ? If yes, insert this information explicitly because the “Summary” leads to some confusion speaking about GKSA related to eq. (10, 12) that are analytical approximations of eqs. (1-4).

 

7. Related to 6) is the discussion on page 6, first paragraph: it tries to explain why the GKSA is much worse than the Bloch equations for short pulses employing eq. (10). Eq. 10, however, is a long pulse approximation of the GKSA from eqs. (1-4). Therefore, the arguments put forward for explanation look not relevant. The authors may provide and improved explanation. It is also not clear in the discussion for short pulses (Figs. 3,4) , why the GKSA provides worse results compared to the Bloch equations than a long pulse analytical approximation (10) of GKSA of eqs. (1-4).

 

Some language improvements:

p.1, Abstract:

….correspondance between the two approaches but…

…theory does not provide satisfactory….

Introduction:

….including ultrashort pulses (USP)…

…a new parameter, namely,…

 

p. 2:

..field in the pulse with carrier….

…Rabi frequency Omega_0 (proportional to the electric field strength) into the…

 

p. 3, second line after eq. (7): space before “Then” is missing

 

p.4, third line after eq. (10)

…makes sense if W £1. It can be seen…

First line after eq. (12): point missing after equation.

 

p. 5, first and 5^th  line:

So we obtain likewise a Lorentz profile, but…

..test the GKSA via the numerical…

 

p. 6,7 Figure captions 1-4:

write explicitly GKSA (eqs. 1-4)-dashed line

write explicitly eq. (12) after saturation

It is not clear, why eq. (10) is not plotted which is a more general analytical approximation that is not really more complex than (12)

In Fig. 4, it might be of interest to plot also eq. (13) of the saturation limit to understand how close s=9 is.

 

p. 8,9, Figures 5,6:

what means Omega_0 in relative units, is it atomic units ?

 

p.9, Summary:

first phrase not clear..: Probably what the authors want to say is

…approach to the probabilistic description of the process based on expression (1) by taking into account the….?

6^th line: write behind GKSA (eqs. 1-4)

11^th line: …Fig. 4, the long pulse analytical approximation with saturation (10) and without saturation effects (12) of GKSA do not give…

 

Author Response

First of all, we would like to express our gratitude to the Reviewer for his/her valuable work. We are confident that corrections based on his/her comments will make our article better.

1. Eq. (1) and following: what units are used ?

We used Gaussian units

2. Concerning eqs. (14-16):

- R1,2,3 are not defined: write in first phrase after Bloch vector the expression (R1,R2,R3)

We add the paragraph with definition of optical Bloch vector (highlighted in yellow after formula (16)).

- write T1 and T2 are the relaxation times of levels 1 and 2, respectively.

It is a misunderstanding. T₁ is population relaxation time of two-level system, it is defined after formula (16). T₂ is phase relaxation time which is connected with spectral width of dipole-allowed transition in two-level system. Parameter T₂ is defined after formula (3).

3. Eq. 17: provide the information in what interval R3 can be, 0-1 ? For consistency, write also about R1, R2.

Corresponding sentence is added after formula (17)

4. Last phrase of paragraph 4: why T1->infinity is correct ? Add short explanation

Corresponding sentence is added (in yellow).

5. Paragraph 5: what means relative units? Relative to what ? Same for all figures captions. Provide information how a value of tau, t1, Omega_0 can be transformed to seconds and angular frequency.

All variables in the paper are in relative units. It is possible because the probabilities (10)–(12) and Bloch equations (14)–(16) are invariant relative to the following transformation:

Frequency variables ® Frequency variables/w_s

Time variables ® Time variables´w_s

Here w_s is some scaling frequency.

For example if w_s=atomic frequency=4.13×10¹⁶ s^-1 then t and all time parameters are in atomic units, i.e. relative time unit=2.4×10^-17 s.

Corresponding sentence is added in the beginning of paragraph 5 (highlighted in yellow).

6. It remains unclear in the figures what GKSA means, is it numerical solution of eqs. (1-4) ? If yes, insert this information explicitly because the “Summary” leads to some confusion speaking about GKSA related to eq. (10, 12) that are analytical approximations of eqs. (1-4).

GKSA means the use of (1) for the excitation probability with spectral profile (3) which account for saturation. This is stated at the end of the 2nd paragraph.

Indeed, (10) is an analytical result for the case of an exponential pulse, which is considered in the article for specifics, i.e. it is a special case of the GKSA. Monochromatic limit (11) reproduces the result of Karplus-Schwinger in terms of the probability of excitation of a two-level system while (12) is excitation probability neglecting the saturation effect but accounting finite duration of the pulse.

In our calculation we used formulas (10) as basic expression and (12)-(13) as specific cases.

7. Related to 6) is the discussion on page 6, first paragraph: it tries to explain why the GKSA is much worse than the Bloch equations for short pulses employing eq. (10). Eq. 10, however, is a long pulse approximation of the GKSA from eqs. (1-4). Therefore, the arguments put forward for explanation look not relevant. The authors may provide and improved explanation. It is also not clear in the discussion for short pulses (Figs. 3,4) , why the GKSA provides worse results compared to the Bloch equations than a long pulse analytical approximation (10) of GKSA of eqs. (1-4).

It is a misunderstanding. Formula (10) is valid for all pulse durations in the framework of GKSA. This formula is not long pulse approximation of GKSA.

Concerning short pulse case (Fig.3, 4). These figures demonstrate the limits of applicability of GKSA. GKSA is model which is natural generalization of Karplus-Schwinger approach on probability description of quantum system excitation. Naturally, the model description has its limits of applicability, which are established in this article. The reasons for the discrepancy between the model and the exact result are not always possible to determine.We think that this cannot be done in considered case.

 Some language improvements:

p.1, Abstract:

….correspondance between the two approaches but…

…theory does not provide satisfactory….

Introduction:

….including ultrashort pulses (USP)…

Done

…a new parameter, namely,…

1. 2:

..field in the pulse with carrier….

Done

…Rabi frequency Omega_0 (proportional to the electric field strength) into the…

1. 3, second line after eq. (7): space before “Then” is missing

Done

 p.4, third line after eq. (10)

…makes sense if W £1. It can be seen…

Done

First line after eq. (12): point missing after equation.

1. 5, first and 5^thline:

So we obtain likewise a Lorentz profile, but…

Done

..test the GKSA via the numerical…

1. 6,7 Figure captions 1-4:

write explicitly GKSA (eqs. 1-4)-dashed line

Done

write explicitly eq. (12) after saturation

Formula (12) is written under the assumption that there is no saturation, as stated in the text of the article before this formula.

It is not clear, why eq. (10) is not plotted which is a more general analytical approximation that is not really more complex than (12)

We use (10) for GKSA calculations

In Fig. 4, it might be of interest to plot also eq. (13) of the saturation limit to understand how close s=9 is.

Calculation shows that in this case the result of (10) coincides with the result of (13). It also can be seen analytically from (10) because W₀ T₂ >>1.

1. 8,9, Figures 5,6:

what means Omega_0 in relative units, is it atomic units ?

This question is explained above

 p.9, Summary:

first phrase not clear..: Probably what the authors want to say is

…approach to the probabilistic description of the process based on expression (1) by taking into account the….?

Done

6^th line: write behind GKSA (eqs. 1-4)

It is misunderstanding

11^th line: …Fig. 4, the long pulse analytical approximation with saturation (10) and without saturation effects (12) of GKSA do not give…

Done

 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The author's addressed all my comments and therefore I can now recommend the acceptance of the paper.