Theory of Time-Resolved Optical Conductivity of Superconductors: Comparing Two Methods for Its Evaluation
Round 1
Reviewer 1 Report
In the paper, the authors theoretically studied the time-resolved optical conductivity for driven superconductors using the nonequilibrium Green’s function method. In particular, they employed two different definitions of the optical conductivity for nonequilibrium systems, depending on the choice of time with respect to which Fourier transformation is performed. Both methods provide qualitatively similar results, including the suppression of the superconducting order after the pump and the Higgs oscillation. On the other hand, these methods have different features in terms of the change before the pump and averaging nature over the current decay time.
The presentation of the results is given in a comprehensive way. The paper clarifies the issue of ambiguity in the definition of the optical conductivity, and proposes a way to improve the time resolution. This is an important progress in this field. Based on these, I can recommend the publication of the paper in Condensed Matter journal. The followings are my comments which I hope the authors will take into account.
1. In Fig. 2, both methods I and II exhibit the overall increase of the optical conductivity over the entire frequencies after the pump excitation. What is an intuitive understanding of this? Does this preserve the sum rule?
2. The functional derivative method to derive nonequilibrium optical conductivity including vertex corrections has been used in an earlier paper, Phys. Rev. Lett. 103, 047403 (2009).
3. While the authors stated that the attribution of the Higgs mode is under heavy debate, there has been a recent progress in the understanding of the importance of impurity effects, which greatly enhances the light-Higgs coupling in dirty superconductors. One can mention recent papers, Phys. Rev. B 99, 224510 (2019), Phys. Rev. B 99, 224511 (2019).
Author Response
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Reviewer 2 Report
This manuscript theoretically studies time-resolved optical conductivity of superconductors, where the authors compare two methods for calculating time-dependent optical conductivities. The authors compare the methods with different treatments about how the probe and gate times are swept to find that the conductivity with the gate time fixed removes the artifacts (response before a pump) in the other method.
The subject-matter is interesting, given recent interests in non-equilibrium responses in condensed matters induced by laser. Specifically, there has been a debate about how to define optical conductivities in ultra-fast situations. The paper is sound and will benefit the audience in this field, but I have some comments as follows:
1. In Introduction, the authors say about a non-equilibrium response in superconductors "that has been attributed to the Higgs amplitude mode of the superconductor [2] (although this subject remains under heavy debate) [3-5]". I am rather surprised that the references ([3-5]) the authors quote are the papers up to 2016, while there is an important development in the subject after that year. Namely, it is now being revealed that the impurity effect is quite important in the Higgs mode in superconductors. A typical experimental system, NbN, is close to the dirty limit (which the present work also adopts), and in such a situation the coupling between the Higgs and light is drastically enhanced, as in the retardation effect in the phonon-mediated interaction. So a generally accepted idea now is that Higgs mode dominates over the quasiparticle contribution, at least in NbN. The relevant papers are
D. M. Kennes et al, Phys. Rev. B 96, 054506 (2017);
T. Jujo, J. Phys. Soc. Jpn. 87, 024704 (2018);
Y. Murotani et al, Phys. Rev. B 99, 224510 (2019);
M. Silaev, Phys. Rev. B 99, 224511 (2019).
There is also a recent review article,
R. Shimano and N. Tsuji, arXiv:1906.09401.
Thus I think the "heavy debate" is not a fair expression any longer. While the present manuscript is on a general methodology, not specifically to the Higgs mode, I think the description should be revised.
2. A key quantity in the present work involves what the authors call "the gate time t_{gate}", but there should be many readers who are not too familiar with the pump-probe setup, especially the "gate time", so the authors might like to explain it in more detail.
3. Related to this, just below Eqn(4) the authors say "there are now three fixed points in time rather than two", but a reader cannot see this from that equation.
4. The present work concludes that Method II is more physical. Then the authors should discuss whether the existing literature in the field mostly use Method I (with the artifact) or there are some papers that evade this point.
Some technical points.
5. In Fig.2, the result is displayed only for \omega > some finite value (~ 0.1). But \sigma_1 and \sigma_2 are Kramers-Kronig related throughout? In the explanation of the same figure, a quantity 2\Delta appears, but \Delta (presumably the BCS gap) is not defined and its numerical value is not quoted.
6. About the non-equilibrium Keldysh formalism, Refs [12,16,18] are cited, but Ref [4] also uses this formalism.
Some minor point.
7. The manuscript is not prepared carefully enough. For instance, in Fig.1, caption, there are two entities explaining panel (c) [(c) Schematic .... (c) Probe currents ...]. The quantity t_{pp} is not defined there (appearing in a later part of the main text). Incidentally, in the same figure, panel (d), top left inset, two curves are displayed where they look identical, but there is a significant difference (at least after a Fourier tr)?
So I conclude that the manuscript should be amended before it is accepted for publication.
Author Response
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Reviewer 3 Report
condensedmatter-554469 - Referee report
(Dated: July 19, 2019)
The manuscript studies theoretically the optical conductivity in superconductors perturbed by an intense pump pulse, which brings the system out of equilibrium and precedes a weaker probe pulse. The latter records the dynamics of the system while restoring the equilibrium regime. The authors make use of the Keldysh formalism to capture the out-of-equilibrium dynamics and consider two distinct methods, I and II as in the text, related to different ways of recording the dynamics of the probe after the pump pulse has passed over. The results corresponding to each method are then compared.
Recently, the pump-probe experimental protocol revealed its potential for investigating symmetry broken phases of matter in which the energy gap between the ground state and the lowest energy excited states ranges in the THz region of the em spectrum. This topic is gaining further interest as both novel experimental techniques and theoretical calculations are looking to provide a detailed interpretation of the physics at play during and after the em perturbation.
I have the following questions and comments:
The method I and II refer to different experimental protocols. In the method I the observation (sampling) time, tgate, is varied and the oscillating current is recorded as a function of tgate. In the method II what is varied is the time delay between the pump and probe pulses by fixing the observation time at a given distance from the pump pulse. While the method I found many experimental examples in the literature, the method II is not so common. Generally, when varying the time delay between the pump and probe pulses, one fixes tgateat a given distance from the probe, as described e.g. in the Ref.s [2] and [19]. In this respect, the method II mentioned by the authors differs from the usual experimental protocols. The authors should report on the experimental relevance of the method II.
The Fig. 1(d) lacks the time units, as well as the caption.
In the section 2-Methods, the authors mention thedirty limit of BCS superconductors and its resulting signal below the energy of the pairing bosons. The authors should add references to this sentence.
For completeness, in the section 2 the authors should write the model Hamiltonian used in the calculations.
In the Table 1: the parameters used for modeling the em pulses lack the energy units. I guess that
the frequencies ωp and ω are expressed in eV, as the other energies in the rest of the table. However, if this is the case, I actually don’t understand what is the utility of having an eV pump pulse and a THz probe one. Generally, the opposite situation in considered, as the frequency of the driving pump field should be in the THz range, to be comparable to the SC gap.
6. What is the SC gap at equilibrium? This is an important parameter of the problem, which deserves to be mentioned at some point of the manuscript. From the Fig. 3, I guess that: 2∆ ≃ 0.083eV, which means ∆ ≃ 10THz. This value is rather high (see the case of the conventional superconductor NbN, with ∆NbN ≃ 0.65THz), and corresponds to a strong coupling limit, as the authors mention in the text. This is not so general. Since the authors state that the method should be general and may be adjusted to simulate realistic materials, I suggest to consider a more conventional system or to justify in detail why the strong-coupling limit is considered here.
7. Rows 65 and 70: I guess that the reference to Figure 1(b) should actually refer to the Fig. 1(d).
8. In the Fig. 2 the authors should show the optical gap at equilibrium, 2∆. In addition, there are missing units in the colormap bar.
9. The oscillations reported in the panels of the Fig. 3 have a typical frequency of approximatively 0.01eV, which is just the frequency of the probe. In my opinion, these oscillations simply trace the linear response to the probe pulse, and not the characteristic Higgs oscillations. The latter should indeed occur at 2∆(∼0.083eV), as reported in the literature and also mentioned in the introduction of the manuscript. The authors should clarify and justify this point. Why are the oscillations at 2∆ not visible? In addition, the panel (a) lacks the units.
While the comments 1-8 represent minor remarks, the comment 9 raises fundamental questions. Consequently, even if I appreciate the framework established by the authors, as well as the numerical efforts that have been dealt with, I actually cannot recommend the present manuscript for publication unless the issues of the comment 9 are satisfactorily addressed.
Comments for author File: Comments.pdf
Author Response
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Reviewer 4 Report
The authors address two schemes to calculate the time-resolved optical conductivity in a pumped superconductor: one is with fixed probe time, another is with fixed gate time. The results are compared and discussed. The disadvantages of Method I can be summarized as follows: the phenomenon of "perturbed free induction", and the limitation on the time resolution, both of which are influenced by the decay rates of probe currents. The argument is sound and the results are interesting. I'm happy to recommend it for publication in Condens.Matter.
Before closing the report, I would like to take the opportunity to make some suggestions for the authors before they submit the revised manuscript for publication.
1. The data in Method II (the vertical cuts in Fig. 1(d)) are built versus Akima spline interpolation on the data obtained in Method I (the horizontal cuts). Is this interpolation always reliable? I'm concerning the case of slow-decay probe currents. The authors may consider adding some comments on this issue.
2. I'm wondering which method is more suitable when going to the comparison with experimental data, and how the results depend on the details of probing pulse? The authors may like to provide additional discussions on it, if possible.
Please note that there seems a typo in the caption of Fig. 1 as: "(c) Probe currents obtained from ...".
Author Response
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Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
In the revised manuscript the authors have taken care of all the points raised by this referee. So I find the revised paper much clearer. However, I still have one point:
In Introduction, the authors have changed the sentence on the Higgs amplitude mode from "although this subject remains under heavy debate [3-5]" into "although on this subject some open questions remain [3-5,12,13]". However, the added references do not claim open questions, but show that the dirty limit (to which the experimental system and also the present work belong) makes the Higgs mode dominant through an enhanced coupling between the mode and light. Thus I don't think that the claim of the open question is an adequate description.
Comments for author File: Comments.tex
Author Response
We thank the referee once again for their careful reading of the manuscript.
We have amended the manuscript to remove the notion of `open questions` as the referee suggested. The new sentence reads:
"...at a frequency of twice the superconducting gap ($2\Delta$) that has been attributed to the Higgs amplitude mode of the superconductor [2-10], although the contribution from light-induced excitation of the Cooper pairs isalso shown to be important[11-13]." Sincerely, A.F. Kemper (for the authors)
