From Symmetry Breaking via Charge Migration to Symmetry Restoration in Electronic Ground and Excited States: Quantum Control on the Attosecond Time Scale
Round 1
Reviewer 1 Report
This manuscript is a reasonable follow-up of the study published in PRL 121, 173201 (2018). In this respect, it would be suitable for publication in its current form.
After giving a new careful reading to the manuscript, and to the previous works of the authors, I should first apologize for assuming that this manuscript was presenting the exact same results as in the PRL 121, 173201 (2018).
As the authors explained, one of the two main differences is that “…. symmetry is restored in the excited state, not in the ground state. “. Looking carefully to the graphs and the labels, it is clear that now the population transfer goes into the excited state, instead of going back into the ground. The same method is applied, the same simplified two-level system is employed to describe excitation in the benzene molecule, but a different combination of pulses is proposed to achieve population transfer into the excited state, instead of the ground.
The authors have pointed out the second main difference “the theory of symmetry restoration in PRL 121, 173201 (2018) is applied to multiple photon excitations in the weak field case, whereas we here apply the model to a strong field scenario”. I can understand that the pulses required now are more intense, since now most of the population should be transferred from the ground into the excited state. However, I still cannot follow where “strong field effects” are being shown. Indeed, with the new figure 2, it seems clear that the authors are only considering a multiphoton perturbative regime.
Nevertheless, I appreciate the introduction of this Fig. 2, where now it is partly justified why these two states are the dominant contributions in the proposed scheme for nuclei frozen at the equilibrium geometry. However, taking into account the actual potential energy surfaces could change this oversimplified energetics, even in the first few femtoseconds.
In my opinion there are three main criticisms to this theoretical work, but it should be noted that they would also apply for the study published in PRL 121, 173201 (2018):
1) There is no novelty in the concepts employed. Analogous control schemes have been explored over the years in more realistic descriptions of atomic and molecular targets. The concepts of “symmetry restoration” or “charge migration” are here employed just to refer to a time-varying superposition of two molecular orbitals.
2) The validity of a simplified two-level system for benzene molecule where the nuclei are assumed to be frozen for at least 10 fs, such that the energies of the electronic states ensure that only these two states are relevant, would require further simulations to be proven. Nowadays, there are a manifold of Quantum Chemistry Packages that would allow to explore potential energy surfaces to check the validity of this scheme.
3) The control scheme involve very well characterized phase-controlled 8 eV pulses with few-femtosecond durations and electromagnetic amplitudes of 106-107 V/cm. I am not aware of well characterized coherent light sources providing these parameters or similar ones.
Author Response
This manuscript is a reasonable follow-up of the study published in PRL 121, 173201 (2018). In this respect, it would be suitable for publication in its current form.
Our response: In this respect, we are of course delighted about this resume.
After giving a new careful reading to the manuscript, and to the previous works of the authors, I should first apologize for assuming that this manuscript was presenting the exact same results as in the PRL 121, 173201 (2018).
Our response: We accept the apology.
As the authors explained, one of the two main differences is that “…. symmetry is restored in the excited state, not in the ground state. “. Looking carefully to the graphs and the labels, it is clear that now the population transfer goes into the excited state, instead of going back into the ground. The same method is applied, the same simplified two-level system is employed to describe excitation in the benzene molecule, but a different combination of pulses is proposed to achieve population transfer into the excited state, instead of the ground.
Our response: We appreciate the referee’s summary of the novelty of our article. We would like to add an important detail and re-emphasize that the irreducible representation of the excited state (IRREPe) should be different from the ground state (IRREPg ). Else there is neither symmetry breaking nor symmetry restoration of electronic structure.
The authors have pointed out the second main difference “the theory of symmetry restoration in PRL 121, 173201 (2018) is applied to multiple photon excitations in the weak field case, whereas we here apply the model to a strong field scenario”. I can understand that the pulses required now are more intense, since now most of the population should be transferred from the ground into the excited state. However, I still cannot follow where “strong field effects” are being shown. Indeed, with the new figure 2, it seems clear that the authors are only considering a multiphoton perturbative regime.
Our response: We have carried out additional quantum dynamics simulations which allow the comparison with the previous results in PRL 121, 173201 (2018) in order to show that the present new scenario for symmetry breaking and symmetry restoration is well beyond the weak field domain. One of the characteristics of the weak field limit is that the population transfer increases linearly with laser intensity. The laser pulses for symmetry breaking and symmetry restoration in PRL 121, 173201 (2018) have weak field strength Eb = Er = 4.42×106 V/cm and achieve 0.0037 population transfer between the ground and excited states. The present laser pulses have much stronger field strengths, Eb = 4.207×107 V/cm and Er = 7.192×107 V/cm. Assuming that they are still in the weak field limit, then the population transfers should be 0.0037×(42.07/4.42)2 = 0.33 and 0.0037×(71.92/4.42)2 = 0.98, respectively. But in reality, the population transfers are 0.30 and 0.70, respectively. The deviations show that while the present laser pulse for symmetry breaking may still be considered in the weak field domain, the required high field strength of the pulse for symmetry restoration is well beyond that limit.
Modification of the manuscript: We have added the following text at the end of the first paragraph of p. 18:”….pulse for symmetry restoration, Er = 7.192×107 V/cm. It is illuminating to compare the population transfers between the electronic ground and excited states, namely 0.30 and 0.70 which are achieved by these pulses, with the value 0.0037 for the weak field strengths Eb = Er = 4.42×106 V/cm that are employed in Ref. [19]. Assuming the weak field limit, the population transfers should increase linearly with intensity, i.e. one would expect the values 0.0037×(42.07/4.42)2 = 0.33 and 0.0037×(42.07/4.42)2 = 0.98. The deviations from the values 0.30 and 0.70 show that while the intensity of the pulse for symmetry breaking may still be considered as “weak”, the required high field strength of the pulse for symmetry restoration is well beyond that limit.”
Nevertheless, I appreciate the introduction of this Fig. 2, where now it is partly justified why these two states are the dominant contributions in the proposed scheme for nuclei frozen at the equilibrium geometry. However, taking into account the actual potential energy surfaces could change this oversimplified energetics, even in the first few femtoseconds.
Our response: See the response to item 2) below.
In my opinion there are three main criticisms to this theoretical work, but it should be noted that they would also apply for the study published in PRL 121, 173201 (2018):
1. 1a) There is no novelty in the concepts employed. Analogous control schemes have been explored over the years in more realistic descriptions of atomic and molecular targets.
Our response: In contrast with the referee’s opinion, we claim that our concept for laser control by a second pulse that is added coherently to the first pulse such that the two pulses add up to a π-pulse is entirely new. The coherent addition of the two pulses requires attosecond precision of the time delay and puts our new concept into the domain of attosecond chemistry. This concept has never been published before.
We would like to add a psychological aspect: Previously, in her/his first report, the referee had claimed that our results have been published before in PRL 121, 173201 (2018). She/he has apologized for her/his mistake, and we have accepted the apology. But now she/he repeats her/his mistake by claiming that “There is no novelty in the concepts employed. Analogous control schemes have been explored over the years…” Essentially, this blames us again for trying to publish a concept which is not new. We claim that the referee will not be able to support her/his reproach by any quotation of the literature, from the fields of coherent control or attosecond chemistry.
1b) The concepts of “symmetry restoration” or “charge migration” are here employed just to refer to a time-varying superposition of two molecular orbitals.
Our response: The general way of expressing this reproach is wrong because it misses the essential key to our concept from symmetry breaking via charge migration to symmetry restoration in electronic ground and excited states, namely the states must have different IRREPs.
2. The validity of a simplified two-level system for benzene molecule where the nuclei are assumed to be frozen for at least 10 fs, such that the energies of the electronic states ensure that only these two states are relevant, would require further simulations to be proven. Nowadays, there are a manifold of Quantum Chemistry Packages that would allow to explore potential energy surfaces to check the validity of this scheme.
Our response: Our article has a new concept of laser control, from symmetry breaking via charge migration to symmetry restoration in ground and excited states. This is demonstrated for an example that we have chosen carefully from the quoted literature. In particular, Ref. [13] has already shown that the vibrational effects on laser induced electronic currents in benzene are negligible during the first ten femtoseconds. We do not agree with the referee’s suggestion to provide additional proofs of the validity of the model which have already been published.
3. The control scheme involve very well characterized phase-controlled 8 eV pulses with few-femtosecond durations and electromagnetic amplitudes of 106-107 V/cm. I am not aware of well characterized coherent light sources providing these parameters or similar ones.
Our response: We agree with the referee, but at the same time we think that it is one of the privileges of theory to predict new phenomena which may well pose challenges to experiment. Moreover, our new concept of laser control from symmetry breaking via charge migration to symmetry restoration is general and should apply to many other molecules – it is easy to predict that the present article will stimulate the search for candidates with more suitable parameters for experimental verification.
Modification of the manuscript: We have added one last sentence at the very end of the Conclusion section: “From the experimental point of view, one should keep in mind that the present new concept of laser control from symmetry breaking via charge migration to symmetry restoration is general and should apply to many other molecules – the present example of the benzene molecule should stimulate the search for candidates with more suitable parameters for experimental verification.”
Reviewer 2 Report
This is an excellent, original manuscript on a very timely topic. The authors propose a new strategy for electronic-symmetry restoration in a pump-probe scheme. The new aspect compared to previous work by the authors is that the probe pulse excites the symmetry-broken superposition state created by the pump pulse to the excited state, thereby restoring the symmetry. Advanced quantum-mechanical calculations on a realistic model system (benzene) are presented. Therefore, the presented study is conceptually attractive and will be particularly well received by the attosecond-spectroscopy community. I therefore recommend publication of this nice study in its present form.
Author Response
@page { margin: 0.79in } blockquote { margin-bottom: 0in; direction: ltr; color: #00000a; text-align: left; orphans: 2; widows: 2 } blockquote.western { font-family: "Liberation Serif", serif; font-size: 12pt; so-language: en-US } blockquote.cjk { font-family: "WenQuanYi Zen Hei"; font-size: 12pt; so-language: zh-CN } blockquote.ctl { font-family: "FreeSans"; font-size: 12pt; so-language: hi-IN } p { margin-bottom: 0.1in; direction: ltr; color: #00000a; line-height: 120%; text-align: left; orphans: 2; widows: 2 } p.western { font-family: "Liberation Serif", serif; font-size: 12pt; so-language: en-US } p.cjk { font-family: "WenQuanYi Zen Hei"; font-size: 12pt; so-language: zh-CN } p.ctl { font-family: "FreeSans"; font-size: 12pt; so-language: hi-IN } Thanks. We are very appreciate about this very positive evaluation.Reviewer 3 Report
The draft has theoretically described a strategy of breaking and restoring of the electronic structure symmetry of molecules using two well-designed laser pulses. Different from the previous work Ref.(19,20), in this draft, the “symmetry” of the molecular will be broken by the first pulse and the molecular is restored to the excite state with the same symmetry group but different irreducible representation after the second pulses (which is no longer the time-reversed copy of the first pulse). This could be understood as that under certain conditions (for example, equ 52 & 53), the population transfer induced by two pulses is equivalent to a resonant pi pulse. This work is very interesting and is an extension of the authors’ previous work Ref(19,20). I feel the readers with ultrafast laser background will benefit from this research. The draft is organized well and the language is fluent and precise. Therefore, I recommend the paper for publication. However, I do have some comments.
1 Some equations are in wrong format. For example, on page 14, equ. 54 and 56 should use cases. And there are also periods in equ. 57 and 59 and so on.
2 Fig. 3 is too small to read. Also, what is the phase in (d), is the phase of the electrical field or the phase of the molecular wave function?
3 The authors have considered the case with an ultrashort pulse which might even be a few femtoseconds. For such a short pulses, I understand it could avoid the unwanted transition due to selection rules, but how about Raman scattering? Is possible that such pulses will induce a resonant Raman scattering? If the intensity is high enough, I think such a pulses might lead to the impulsive stimulated Raman scattering for molecules as well. If this is true, it might be very difficult to study it experimentally.
Author Response
@page { margin: 0.79in } p { margin-bottom: 0.1in; direction: ltr; color: #00000a; line-height: 120%; text-align: left; orphans: 2; widows: 2 } p.western { font-family: "Liberation Serif", serif; font-size: 12pt; so-language: en-US } p.cjk { font-family: "WenQuanYi Zen Hei"; font-size: 12pt; so-language: zh-CN } p.ctl { font-family: "FreeSans"; font-size: 12pt; so-language: hi-IN }The draft has theoretically described a strategy of breaking and restoring of the electronic structure symmetry of molecules using two well-designed laser pulses. Different from the previous work Ref.(19,20), in this draft, the “symmetry” of the molecular will be broken by the first pulse and the molecular is restored to the excite state with the same symmetry group but different irreducible representation after the second pulses (which is no longer the time-reversed copy of the first pulse). This could be understood as that under certain conditions (for example, eqn. 52 & 53), the population transfer induced by two pulses is equivalent to a resonant pi pulse. This work is very interesting and is an extension of the authors’ previous work Ref(19,20). I feel the readers with ultrafast laser background will benefit from this research. The draft is organized well and the language is fluent and precise. Therefore, I recommend the paper for publication.
Our response: We are delighted about the overall very positive evaluation of our contribution.
However, I do have some comments.
1 Some equations are in wrong format. For example, on page 14, equ. 54 and 56 should use cases. And there are also periods in equ. 57 and 59 and so on.
Our response, and modification of the manuscript: We have corrected the wrong formats of the equations 54, 56, 57, 59 and have carefully searched for, and corrected analogous wrong formats in other equations.
2 Fig. 3 is too small to read. Also, what is the phase in (d), is the phase of the electrical field or the phase of the molecular wave function?
Our response, and modification of the manuscript: We have enlarged the format of Figure 3. In the Figure legend 3(d), we have added: “Panel (d): Phase difference…of the wave functions in electronic excited and ground states…”
3 The authors have considered the case with an ultrashort pulse which might even be a few femtoseconds. For such a short pulses, I understand it could avoid the unwanted transition due to selection rules, but how about Raman scattering? Is possible that such pulses will induce a resonant Raman scattering? If the intensity is high enough, I think such a pulses might lead to the impulsive stimulated Raman scattering for molecules as well. If this is true, it might be very difficult to study it experimentally.
Our response: Stimulated Raman scattering implicitly assumes excitation – de-excitation by a two-photon processes. These terms are appropriate for the domain of sufficiently weak fields - in particular one may apply them for describing the processes during the scenario of the previous PRL 121, 173201 (2018). The present new concept from symmetry breaking via charge migration to symmetry restoration in ground and excited states calls for stronger laser pulses – see our detailed reply to the introductory comment of Referee 1. We agree with Referee 2 that in this domain, it becomes exceedingly difficult and problematic to analyze the details and to unravel the effects of the laser pulses in terms of few photon processes such as Raman scattering. But the lack of easy interpretations in terms of few-photon processes would of course not block the experimental application.
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
In this work the authors elaborate a theory for symmetry breaking and symmetry restoration
in finite systems with a discrete spectrum.
The theory, however, is too oversimplified since it assumes that only two many-body quantum states are populated during the time evolution. There are several problems with this assumption
1) The dipole operator connects the ground state g to a single excited state e, for otherwise the evolution operator in Eq. (28) would not be block diagonal. However, in reality this is not the case. Why this form of the dipole operator is physically relevant?
2) Selectivity could also be realized through energy matching. However, the external field is a pulse and hence the photons have an energy distribution. Consequently, there will be several excited states involved in the time propagation.
3) A remark. The Hamiltonian is not constant in time when the laser is active. Therefore the evolution operator does not depend on the time difference as the notation in page 6 suggests (and as it is explicitly stated below Eq. (28)).
4) Even assuming that the simplification of the authors is relevant to some realistic physical situation the resulting theory is a theory for a two-level system which can be solved exactly by several means. It is not clear what the theory of the authors adds to what is already known. In particular, it would be helpful for the reader to understand the connection, if any, with the following observation. If the Hamiltonian H(t) = -H(-t)+identity*f(t) (where f is an arbitrary function of time) then the wavefunction satisfies psi(t)=phase(t)*psi(-t).
Before I can recommend this paper for publication the authors should clarify the relevance of their theory by addressing points 1) to 4).
Reviewer 2 Report
The manuscript entitled “From symmetry breaking via charge migration to symmetry restoration in electronic ground and excited states: Quantum Control on the attosecond time scale” presents a theoretical study on the possibility of breaking and restoring the electronic structure symmetry in a simplified two-level model system by using two laser pulses. The manuscript is well written and presented. However, I have some serious concerns regarding the novelty of this study and therefore I do not recommend for publication in the present form.
The methodology and all the results here presented have been previously published in PRL, 121, 173201 (2018) by the same authors. This reference is provided and properly cited in the manuscript (reference 19). However, to my understanding, there is no novel data or new progress on methodology with respect to reference 19. The method, which is only the analytical solution of the TDSE for a two-level system, has been described in detailed in the supplementary material of ref. 19. Therefore it is difficult to justify the originality/novelty of this manuscript for publication. Even Figure 2 in the current manuscript is the exact same figure as Fig 2 (including the upper panel of Fig1a) in ref. 19. Therefore, I cannot recommend this manuscript for publication, because it has been published already, unless the authors can justify it or/and include additional insights/data/discussion on the topic.
Besides, there is a number of other issues that should be addressed:
- While the observations and conclusions withdrawn from this simplified theoretical approach, limited to two-level systems, could provide a quite realistic description of this phenomenon in specific atomic targets (where the energetics allows it), it is more difficult to assess its appropriateness for molecular targets . Indeed, could the authors confirm if it would be realistic to transfer all population from the ground state of benzene molecule to its excited state E1u? There are existing works on photoexcitation of benzene (see, for instance, Chem. Soc. Rev. 2015,44, 6472-6493 and references there in). Is there any proof that a complete population transfer would be possible using such broad pulses? The authors should provide, at least, the energetics of the lowest excited states of benzene, and proof that the broad bandwidth of the pulses employ only excite the only excited state under consideration.
More importantly, there are some recent works by M. Robb and collaborators on ionized benzene, where they show that nuclear coherence is lost in just a few femtoseconds, therefore assuming that only two states are relevant (ignoring possible conical intersections), and even more, that the population will be fully transfer to one of them is unlikely, unless the nuclear wave packets associated to the states populated upon the first pulse remains unchanged in the 4-6 fs that the second pulse takes to reach the target. Could the authors provide the potential energy curves of the relevant states for benzene molecule and confirm that i) the evolving nuclear wave packet that would be created from the ground state into the excited state remains mostly unchanged during the first 4-6 fs, and that ii) no conical intersections or other phenomena associated to coupled electron-nuclear motion would play a role?
- The authors mention “Since the superposition state is not an eigenstate, it is of course non-stationary, representing charge migration”. This statement is partially true. A non-stationary state does not necessarily lead to observation of charge migration. A wave packet created by two (or more) eigenstates which do not spatially overlap among them does not lead to charge migration. Spatial overlap is a requirement.
- Page 3, line 47: “…authors did not recognize the phenomenon”: it is a strong statement that may not be true. The authors focused on demonstrating the possibility of population transfer, which consequently lead to a superposition of eigenstates and, for specific scenarios (as shown in the simplified two-level systems in this study) can lead to an observable symmetry breaking and restoration. That does not imply necessary at all that the authors are not aware of one of the consequences of the population transfer. Therefore, I would suggest a fair statement such that “the authors do not discuss this phenomenon” or “the authors do not explore the consequence…”.
- Two of the three most relevant references provided as existing works by the same authors and, more importantly, as references employed to present the previously published parts of the method are wrongly cited (refs. 20 and 21):
Reference 20 is indicated with pp. xxx-yyy and Reference 21, which is critical to follow the manuscript is wrong: It does not exist a manuscript published with those authors/title, it does not exist a Faraday Discussions volume 312 in 2018, and the DOI belongs to a different manuscript/authors. It is not possibly to follow what the authors mean in several parts of the manuscript. These are just some examples:
(page 1) “… one can employ a well-designed second laser pulse that restores symmetry [19-21]”.
(page 2) “… previous approaches …. have been demonstrated for two model systems, the Rb atom [19] and the oriented benzene molecule [19-21]”.
(page 2) “… For example, … superposition of benzene had Cs [19-21] or C2v [20] symmetry….. ….. representing charge migration [19-21].
(page 14) “The pulse thus serves as “general” example, quite different from the special p/2 pulse that has been employed in Ref. 21.
- Page 1, line 15: “… we have shown…”, “it has been shown”… (only 4 references over 18 correspond to publications by the authors). It seems unnecessarily misleading. Their own references are properly highlighted later. This first sentence in the introduction together with the references provided has been excessively compacted. It provides references for three distinct (although obviously connected) phenomena:
i) broken symmetry: The references include very different type of studies in excitation (superposition of gerade/ungerade states in bound systems) and ionization (i.e., electron localization in photoelectron emission problems), consequently, very distinct phenomena are explored.
ii) charge migration: which indeed, as the authors show, do not necessarily break the total symmetry. It depends on the electron density associated to the states involved in the wave packet.
iii) restoring symmetry: which, in the present scenario, is nothing else but moving from a non-stationary to a stationary state of the system.
- Typos:
Page 2, 7th line in the paragraph that starts with “The previous approaches…”: missing closing bracket in “ (e.g. due to nuclear motions”.
Page 6, equation (25) should be corrected. The authors explain that the time evolution operators depend on just the time difference, but it depends on the picture employed (Schrödinger or interaction picture).
