On the Husimi Version of the Classical Limit of Quantum Correlation Functions

Round 1

Reviewer 1 Report

This paper compares calculations for the dynamics of a one dimensional Morse oscillator using several linearized semiclassical methods.

I recommend that this paper be published after the authors address a few comments.

It would be much easier to judge the accuracy of the results in figures 1, 4, and 5, if the y axis only covers the range 0 to 0.2. In the discussion of figure 1 on p. 13, the comment is made that the HK result “is not coming back to unity fully at the revival time, which is around t = 75.” The blue line in the figure is a split operator solution of the time dependent Schrodinger equation. Doesn’t this method give the quantum result? This does not show this revival either.  There should be some discussion of why this is so. This paper only treats a one dimensional problem numerically. How does the calculation time for the methods increase with dimensionality?

Author Response

We thank the Reviewer for the helpful comments and suggestions.

We have now included insets to Figs. 4 and 5 with a blow up of the
initial part of the time series. Fig. 1 was corrupted during the
editing process. We have now included the correct Fig. 1, which
shows the full revival! Thanks for realizing that something was
wrong with the ms at this point!

Regarding the scaling with dimensionality we have some experience
with 1D versus 2D which are both shown in the ms. The 2D calculations
do not need more trajectories than the 1D ones in the present case,
but we are unable to give a general mathematical proof for that.

Reviewer 2 Report

In the manuscript, the authors adopt the Herman-Kluk (HK) propagator for calculation of the survival probability in the semiclassical approximation. In contrast to linearized semiclassical approaches, the proposed approach can reproduce the interference phenomena, for example, quantum revivals.

The main question to the manuscript: what is the novelty? The HK propagator was known and was used to describe the interference phenomena in the semiclassical approximation. So, what is the novelty of the presented manuscript? If the authors just take a known method and apply it to yet another calculation, then it doesn't seem significant enough for publication. The authors should provide more explanations in the introduction to clarify this issue. After this, the manuscript should be reconsidered. 

Other questions and comments:

The abbreviations SC-IVR (often usage in the manuscript), TDSE (p.13, line 127), and FFT (caption of Fig.1) should be defined. Though the abbreviation HK is defined on p.9, it should be defined right after the first usage on p.8. Line 59: the survival probability is equal to the auto-correlation function, not to the squared absolute value of it. Eq.(4): What is the definition of p_t and q_t? Do they correspond to the classical trajectory? P.3, line 78: what does it mean that an operator "does not contribute to the phase"? Why does it matter? Eq.(7): probably, H should be replaced by \hat H. A derivation of formula (8) should be provided. P.6, lines 84-86: it is worthwhile to note that the Wighner-Weyl version of the correlation function is equal to the approximate Husimi version (high-temperature, or small beta limit, see the line after Eq.(13)). Eqs.(17) and (21): in some places the small psi is written and in some places the capital Psi is written. What is the difference? What is the definition of the capital Psi? What is the precise definition of the small psi? P.8, line 91: I would say that the Wigner-Weyl version has an additional factor 4, not 2, with respect to the Husimi version, since, in the latter, we have a factor 2 in the denominator. Lines 93-94: the last sentence of the paragraph is totally unclear. I am not sure that I understood what is the problem with the Husimi result. Yes, it differs from the Wigner-Weyl result, but why do we consider the Wigner-Weyl result as the benchmark? Is the problem that the Husimi result does not give unity at the initial time? Why the method presented in Sec.3 can be associated with the Husimi transformation? Yes, it also uses the coherent states, but, for example, Eq.(29) contains single integration over dqdp, but Eq.(35) (if we substitute Eq.(33) in it) has two integrations over dqdp. How can be explained that the method of Sec.3 allows to overcome the limitations of the "direct" application of the Husimi approach? Figs.1 and 5 coincide. Fig.1 does not correspond to the description in Sec.4.1 (no revival, no coincidence for short times with Fig.2).

Author Response

We thank the Reviewer for the helpful comments and suggestions
and will answer the question regarding the novelty below.

We have now spelled out the acronyms used, thanks for pointing
out the omission. Regarding the autocorrelation and its relation
to the staying or survival probability: The autocorrelation is a
quantity with real and imaginary part and therefore cannot be a probability.
The probability is calculated by taking the absolute
value squared.

Furthermore, we state now explicitely that (p_t,q_t) are solutions
to Hamilton's eqns right after Eq. (2) as well as Eq. (34)!
The new finding in our manuscript is the fact that we have
given a Husimi expression for the survival probability.
This case was not covered in a previous version of classical
Husimi approaches, as stated explicitely in the ms (top of page 2) with a new
reference to the already referenced ms by Antipov et al (old and new Ref. 15).

The missing operator hats on H were added, whereas a derivation of Eqn. (8)
is not appropriate as this is a definition. We added a new reference to old Ref. 2,
where this definition is also used. A new statement has been added
to line 85, that the results are identical in the high temperature limit.

Thanks for pointing out inconsistences that occured by the use of lower case psi.
We have now changed all occurences to capital Psi. In addition we
have added references to  Eqs. (27) and (37) (new equation numbers in
new ms because some previously unnumbered eqns. are now numbered!) in
order to make clear what is compared. First we compare just the transforms
of the operators and in the end we make a statement on the survival
probability itself. We hope that now things are clear.

Regarding the relation between old Eq. (29) and old Eq. (35) (now Eq. (39)
and Eq. (47)): the key point is that we could derive a formula with only
a single phase space integral also for the survival probability. So what should
be compared is new Eq. (39) with new Eq. (76), which are both single
phase space integrals.

Thanks for pointing out that two figures coincide. We have now
solved  this issue.

Reviewer 3 Report

This paper provides some interesting theoretical work on extending the Husimi representation version of the linearized semiclassical initial value representation approach (LSC-IVR) to the calculation of survival probabilities.  The Husimi representation version of this method is an efficient and useful approach, but until the present contribution it was somewhat restricted in the type of operators that could be treated.  Here, a formulation that allows the important class of projection operators to be included.  

 

The authors develop their formalism and methodology for treating survival probabilities and compare with Wigner LSC-IVR and the full SC-IVR method by presenting numerical results for the Morse oscillator, a simple but enharmonic problem, as well as the Morse coupled to a one mode harmonic bath.  They illustrate the limitations of the existing Husimi approach, and highlight how their alternative derivation leads to additional exponential terms that are essential to bring the approach into close agreement with the full HK approach.  The paper provides ample details on the derivation and analytical techniques employed, as well as the numerical calculations.  Although highly technical, this paper raises an important point about semiclassical methodology, and provides a substantial step forward in the tools for simulating complex quantum systems.  The paper appears to be scientifically correct, and is a significant contribution to the literature.  It is of interest to the readers of the journal.  

 

One point that the authors neglect to emphasize sufficiently is that the variables q_t and p_t appearing in their correlation functions are *classical* trajectories.  This is the essence of the approximation of using “classical input” in doing quantum dynamics.  The authors should mention this explicitly and it would also be a good idea to discuss the details of how they are computed—-for instance, state that they integrate Hamilton’s equations derived from the Weyl symbol corresponding to the Hamiltonian operator, etc.  Also, it would be helpful to discuss the nature of this approximation in general and if full HK (for instance) can eliminate all errors introduced by classical trajectories or if the full set of ALL trajectories that underlies the path integral foundations must be employed.

 

This paper is acceptable for publication after the authors make minor revisions based on the comments above.  

Author Response

We thank the referee for the helpful comments and suggestions.

We have now stated explicitely that (p_t,q_t) are solutions
to Hamilton's eqns right after Eq. (2) as well as Eq. (34)!
Furthermore, we have added a paragraph on the usage of the
classical Hamiltonian (which is here equal to the Weyl symbol
of the Hamilton operator) in Section 4. Also, at the beginning
of Section 3 a comment on the applicability of the HK propagator
is made.

Round 2

Reviewer 1 Report

The paper is now acceptable for publication.

Reviewer 3 Report

There paper is acceptable for publication.