Fingerprints of hot-phonon physics in time-resolved correlated quantum lattice dynamics

Dear Editor,
thank you very much for having sent us the Referee reports of our paper:
"Fingerprints of hot-phonon physics in time-resolved correlated quantum lattice dynamics",
submitted for publication in SciPost Physics.

We thank also both of the Referees for their positive feedback and
for their valuable comments and recommendations, which help us in improving our manuscript.
We have revised the manuscript along the Referees suggestions
and we think that the revised version is now suitable for
publication in SciPost Physics.

We provide a detailed reply to the Referee comments
as well as a list of the modifications in the reply
to each of the Referee's questions (also attached below).


Thank you very much,

Emmanuele Cappelluti and Dino Novko

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Reply to Referee 1

We are greatful to the Referee 1 for his/her valuable comments
and for the positive assessment of our manuscript.

Below, we provide an answer to each of his/her question.

> 1. Although the authors claim that their analysis is generic,
> but it is not obvious for me if the observed anomaly
> in the lattice displacement correlation factor is a unique fingerprint
> of the hot phonons. It seems to be just a signature of the "out-of-phase
> in-plane motion of the two boron atoms" in MgB2. Would it be possible
> to observe similar features in any system with several atoms in the unit cell,
> if their optical modes are excited? Or would the same measure work
> in other systems with different symmetries of the hot phonon modes?

We thank the Referee for the comment.
Concerning the question whether
"the observed anomaly in the lattice
displacement correlation factor is a unique fingerprint of the hot phonons.
It seems to be just a signature of the "out-of-phase in-plane motion
of the two boron atoms" in MgB2,
we definitively answer: yes.
"Out-of-phase in-plane motion of the two boron atoms" is an extensive
way for defining the E_2g phonon branch,
and the sudden dominance of these modes with respect to other ones
is precisely the hot-phonon physics we are describing.
We also confirm that, as stressed in the end of Sec. 1 of the manuscript,
the scenario investigated in this paper that is focused on MgB2, is quite
general and it can be applied to other systems, with several atoms in the
unit cell and with different symmetries of the hot phonon modes.
The clearest example could be underdoped cuprates [e.g., La_{2-x}Sr_xCuO_4]
where hot phonons have been inferred from a three-temperature
modelling of time-resolved x-ray diffraction [Mansart et al;,
PRB 88, 054507 (2013)]. The symmetries of the relevant (hot) modes
are shown in Fig. 12 of that paper, involving modulation
of correlated motion of bond distances O-O, La-La, La-O etc.
The possibility of probing the correlated motion of different pairs
provides more channels for actual detections of hot phonons.
Other materials (e.g., semiconducting systems such as GaAs) can be
also investigated.
In order to clarify in a better way this point, we have added
few useful references concerning hot-phonon physics
in cuprates in the end of Sec. 1 and we added
a detailed discussion, and a new figure,
about the perspectives in cuprates and other materials
in a new Section (Sec. 4 of the revised manuscript).

> 2. I think the experimental feasibility of detecting the anomalous
> behavior in the correlation factor should be discussed.

We thank the Referee also about this point.
Considering MgB2, which is the template compound
considered in this manuscript, efficient probes of the interatomic
correlations has been provided in literature, at the steady level,
by means of neutron diffraction, which is however not particularly
suitable for femtosecond resolution.
The experimental landscape is however changing day-by-day
and new experimental techniques are developed,
opening new promising pathways.
We discuss the experimental perspectives also
in the new Section 4.

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Reply to Referee 2

We thank also Referee 2 for her/his careful reading
and for the useful suggestions, which we have tried
to incorporate in the manuscript as best as we can.

We provide below a reply for each point with the corresponding changes.

> 1. I would ask the authors to write explicitly at the end of the Introduction
> that Sec. 2 is basically an extended explanation of part of Ref. [36],
> as well as to add explicitly in the captions of Figs. 1 and 2 which figures
> are adapted from Ref. [36].

We agree with the Referee.
Sec. 2 is mainly an overview (although with some novel elements,
like the direct comparison between 2T and 3T models) of former
Ref. [36] (current Ref. [37] in the revised manuscript).
As the Referee understand, this summary was needed and useful in order
to present the basic tools for the new analysis provided in the paper
and for making the manuscript self-contained.
We however agree with the Referee that a most clear stress
that this Section has a strong overlap with Ref. [37] is due.
We have accordingly introduced a new paragraph at the end of the
Introduction, with the scheme of the present work,
clarifying that Sec. 2 presents theoretical modelling from Ref. [37].
We have stressed in the caption of Figs. 1 and 2 that
they are a re-adaptation from Ref. [37].

> 2. About Section 3 - Time-resolved lattice dynamics. The authors introduce
> here the evaluation of the (projected) mean-square lattice displacements
> relative to the phonon modes following Refs. [68,69]. To this end,
> they explain Eq. (12) and show Fig. 3. Here, the phonon statistics
> enters via a Bose-Einstein distribution evaluated at an effective
> lattice temperature: this approximates the non- thermal phonon distribution
> that we expect out of equilibrium. Actually, a very similar analysis
> was carried out in Ref. [R1] (see below), which is not cited
> in the manuscript. Have a look in particular at Eqs. (3), (4) and (5)
> of that reference, as well as Fig. 4 and the animation in their
> supplementary material. The only difference is that the authors
> of Ref. [R1] use an approximated out-of- equilibrium and
> time-dependent non-thermal distribution computed directly
> from first-principles. I would ask the authors to cite Ref. [R1],
> and - if they wish so - to add a sentence in order to clarify
> for the reader the advantages of their approach in the present context.
> [R1] Tong and Bernardi, Phys. Rev. Research 3, 023072 (2021)

We thank the Referee for poiting out to us such interesting reference
that, honestly speaking, we overlooked.
The approach described there, as well in other similar papers,
is very promising in a compelling modelling of out-of-equilibrium time
dynamics avoiding the assumption of effective temperature.
In our present manuscript this level of accuracy is somehow not
necessary, and, for sake of simplicity,
we employ the simplest (effective-temperature based)
model that captures the relevant physics.
We comment on the perspectives of the approach of Ref. [R1]
(presently Ref. [34]) as well as of similar papers,
in a new paragraph in the Sect. 2.

> 3. In Ref. [69], the interatomic correlation factors B-B and B-Mg
> are found to have a value around or below 0.1 in a temperature
> range from 0 to 600 K (Fig. 5c and d of that reference). In Fig. 4b
> of the present manuscript, the values corresponding to the same
> quantities are sensibly higher (0.225- 0.275) both at negative
> time delay and at 0.4 ps, where according to Fig. 2a the
> temperatures
> should be below 500 K. Can the authors clarify this?

We thank the Referee for this comment that allows us to clarify
a subtle point.
The Refeee is right in noticing such discrepancy in the absolute value.
On this regard one should keep in mind that other scattering sources
different than the electron-phonon coupling (more particularly:
disorder) might play a role in the experimental analysis,
affecting the quantitative estimate of the correlation functions
under steady conditions.
The presence of disorder is expected to lead to an overall
offshifts of the value of the correlation factors, not altering
the sign and the presence of hot-phonon-induced anomaly
on which we focus in this paper.
In order not to affect the readibility of the manuscript,
we have discussed in detail these aspects in a long note (Ref. [83]).

> 4. I think the authors may be more specific about the generalization
> of their approach (in this, I agree with point 1. of the first Referee).
> In my opinion, a generalization of the proposed approach may be
> schematized in this way: (1) we know or suspect some specific
> symmetry constraints on the hot-phonon modes of some system
> (in this case, MgB2 hot-phonon modes are characterized
> by out-of-phase boron planar oscillations);
> (2) we devise a suitable correlation factor between atomic displacements
> which is sensitive to such symmetries (in the present case,
> atomic pair projection of B-B relative displacements);
> (3) we check that at equilibrium / two-temperature level there is
> no sensible effect; (4) finally, we investigate the changes at the
> out-of-equilibrium / three-temperature level: if there is
> a marked change in correlation factor (in the MgB2 case, the B-B factor drops),
> then we may conclude that a hot- phonon scenario is likely.
> Do the authors agree with this step-by-step summary?

We confirm that the scheme of the Referee is essentially the scheme
we have in mind.
We thank the Referee for such feedback.
Often something that appears obvious to the authors, is not to obvious
for the reader.
Following the suggestion of Referee 1, we have thus gladly
explicitly summarized such scheme in the second paragraph
of new Sect. 4.

>5. If the authors agree with the scheme in point 3: can they make
> another example of a correlation factor not involving out-of-phase
> oscillations of a specific atom but some other symmetry?

The Referee touches an interesting point.
Although from the theoretical point of view, hot phonons not related
to out-of-phase lattice displacements are certainly possible,
from the point of view of material science the conditions
to observe them might be more difficult.
On the one hand, favourable conditions for hot phonons are usually
realized for the zone-center optical modes, which intrinsically involve
opposite displacement for inner atoms in a unit cell.
However, it is still possible in principle to devise systems
with many atoms in a unit cell where the out-of-phase motion
of a subset of atoms is accompanied by an in-phase motion
of another subset of atoms.
Other possible conditions can be encountered when hot phonons
are stored in the edge-zone modes. Finally, one should consider the
possibility where a large electron-phonon coupling, along
with possible nesting conditions, induces a strong Kohn anomaly
on the acoustic branches, where also non-thermal populations
can be sustained.
Although such complex investigation goes beyond the aims
of the present work, the search for real systems where hot phonons
are associated with an in-phase lattice motion is thus an interesting subject
that we hope can be further developed in the future.
We have briefly commented about these aspects in Sect. 4.

> 6. I think that the authors should discuss a bit more
> the possible limitations of this approach. For example,
> the case of a correlation factor failing to "decrease", while
> hot-phonon behavior is clearly observed by the three-temperature model
> or experimentally. Or, the correlation factor "decreasing" without
> hot-phonon behavior being observed or while the two- and
> three-temperature models give similar results.

We have followed the Referee's advise, smoothing some strong statements, e.g.
"we suggest that a smoking gun evidence" -> "we suggest that a suitable evidence";
"it provides a striking tool" -> "it provides a useful tool".
We agree that a theoretical prediction is always subject to a possible
failure, because of additional and/or spurious effects,
but it is hard at this point to comment about possible failures
before they occur.
We hope that our manuscript can provide a useful guide to
experimental groups for a direct check, and we are at the moment
confident that the anomalies in the correlated motion
here predicted can be revealed.

> 7. I am curious whether strong nonadiabatic effects,
> as in the case of MgB2, would be detrimental to the quantitative accuracy
> of the proposed method, since the DFPT phonon normal modes become
> ill-defined in the presence of these effects, probably even more
> so out of equilibrium. What do the authors think?
> 8. Also, the analysis of the interatomic correlations rests on the
> harmonic approximation for the phonon normal modes, and on the fact
> that they remain harmonic out-of-equilibrium. The authors may want to
> comment on this.

We thank the Referee for arising these points.
As the Referee might have noticed from the list of our previous publications,
the role of nonadiabaticity is a crucial topic that is within the core
of the main interests of the both authors.
MgB2 presents an undoubtful degree of nonadiabaticity which
is intrinsically entangled with anharmoncity.
We therefore completely agree with the Referee that investigating
the interplay between nonadiabaticity and the quantum lattice
dynamics (including anharmoncity), also within the context of hot phonons,
is a crucial task both for a quantitative description of the effects
of the electron-phonon coupling as well as a suitable path
for revealing nonadiabatic effects.
This is however a formidable and delicate task where no well-defined
procedure is assessed in the scientific community so that
different approches have been advanced according different aims,
leading sometimes to controversial debate.
For these reasons, and since nonadiabaticity and anharmonicity
are not fundamental ingredients of the analysis here presented,
in this work we didn't address explicitly the inclusion
of nonadiabatic/anharmonic effects.
A proper inclusion of nonadiabatic/anharmonic effects is however a topic
that we (and hopefully even other researchers)
will certainly pursue in future works.

> 9. Fig. 2(c) introduces the electron-phonon coupling strength
> lambda but this quantity is never defined.

We thank the Referee for pointing out this lack.
We have added the definition of lambda in the present Eq. (1)

> 10. In Eq. (12) the phonon distribution is denoted as n. I just
> assumed that this is the Bose-Einstein distribution,
> which however appears as b in Eqs. (5-6).
> Can the authors solve this discrepancy?

It was a typo. We thank the Referee.
We have used the unique notation b(x) for the Bose-Einstein factor
all through the paper.

> 11. At the bottom of page 8, the authors write
> "... with a sharp increase of rho_{B-B} ...".
> I assume they mean "decrease", considering Fig. 4(b)?

We thank the Referee for having signalized us also
this typo, which we have properly corrected.

> 12. Both in the abstract and in the conclusion, the authors qualify
> their approach as a 'quantum-field-theory approach'.
> I do not understand why. I would qualify this approach a
> 'semiclassical' instead. First, the three-temperature model seems
> to me a semiclassical approach: Boltzmann-type equations
> using DFT-computed parameters (therefore at an assumed
> effective thermodynamic equilibrium), and in the
> relaxation time approximation.
> Second, the correlation functions between DFPT-computed
> atomic displacements are also expectation values on quantum
> harmonic oscillators in effective equilibrium, which may certainly
> correspond to the diagonal elements of an approximated equilibrium
> phonon Green's function, but still I wouldn't call this quantum field theory.

We agree with the Referee that the expression
"quantum-field-theory-approach" was somehow misleading
and, following the Referee's advice,
we have corrected it as "semiclassical approach"