# Absence of quantum features in sideband asymmetry

### Submission summary

 As Contributors: João Machado Preprint link: scipost_201909_00003v5 Date submitted: 2020-01-09 01:00 Submitted by: Machado, João Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Experiment Quantum Physics Approach: Theoretical

### Abstract

Sideband asymmetry in cavity optomechanics has been explained by particle creation and annihilation processes, which bestow an amplitude proportional to 'n+1' and 'n' excitations to each of the respective sidebands. We discuss the issues with this interpretation and why a proper quantum description of the measurement should not display such imbalance. Considering the case of linearly coupled resonators, we find that the asymmetry arises from the backaction caused by the probe and the cooling lasers.

###### Current status:
Editor-in-charge assigned

### List of changes

List of changes:

» The Fourier coefficients of the optical response function in Eq.8 and the optical response for the case where each sideband is driven separately are now displayed in Appendix A.

» The multi-tone case was moved to a separate appendix (App.B)

» Briefly discussed the photon counting detection experiments

- R. Riedinger, S. Hong, R. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gr\"{o}blacher, {\it Non-classical correlations between single photons and phonons from a mechanical oscillator}, Nature {\bf 530}, 313 (2016), \doi{10.1038/nature16536}

- J. Cohen, S. Meenehan, G. MacCabe, , S. Gr\"{o}blacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, {\it Phonon counting and intensity interferometry of a nanomechanical resonator}, Nature {\bf 520}, 522 (2015), \doi{10.1038/nature14349}

in section 4 (conclusion). It now reads:

Further, our analysis is focused only on the power spectral density obtained with linear detections methods. There are other methods closely related to SA which also make use of the term sideband asymmetry'', such as the difference between the photon count rates when an optomechanical system is driven with short pulses at the red and blue sidebands [CITATION], that are outside the domain of the present work. Because single photon detection is a method that does not exhibit the operator order issues raised in Sec.2, and because only the instantaneous count rate is measured instead of the frequency spectrum (for weak driving and short enough pulses, the effects of backaction and noise are negligible), our conclusions do not apply to these single photon detection methods. The quantum description of the photon count detection scheme does indeed display a $n/(n+1)$ imbalance and can be used to directly determine the phonon occupancy. Nevertheless, it would be interesting to understand how the two methods are related to each other.

» The new structure of the article was updated in the introduction:

In this article, we start by discussing the problems with the standard interpretation of SA, their connection to how measurements are performed, and the role of ZPM in the spectrum (Sec. 2). We proceed to compute the response function for a system composed by two driven optical modes and a mechanical one (Sec. 3). Considering the symmetric noise power spectral density for both cases, we show that ZPM does not contribute to the asymmetry and that SA naturally arises from the backaction between the cavity and the mechanical oscillator. Additional details of the calculations can be found in App. A. For comparison with other measurement methods, we also analyse the case of a system composed by two modes (cavity + mechanical), but driven with multiple tones in App. B. Finally, we present a concise review of previous analyses of SA in App. C.

### Submission & Refereeing History

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Resubmission scipost_201909_00003v5 on 9 January 2020
Submission scipost_201909_00003v1 on 7 September 2019

## Reports on this Submission

### Report

As previously stated, there has been prior work on the necessity of considering the symmetrized noise spectral density in order to faithfully describe measurement outcomes (e.g., Ref 6). It is even clearly stated in the standard RMP in the field (see Ref 14). There have also been calculations of optical spectra based upon the normally-ordered first-order correlation function (e.g., Ref 11), as measured via photodetection. While it is accepted that this distinction is not always clearly made, it clearly has been previously discussed at length, particularly in the context of sideband asymmetry.

Therefore, the text around Eq (1) appears to still be largely incorrect.

Also as previously stated, Eq. (1) is still just the Fourier transform of a position correlation function; whether this quantity is interpreted as a measured spectral density is a separate matter. Further, Eq (1) confusingly conflates the calculation of spectra of scattered optical fields with the calculation of a noise spectrum of the mechanical oscillator itself.

I accept Eq (2) though it is misleading to represent this as a novel observation. It would also be helpful to make it clear that (X(\omega))^\dagger and X(-\omega) are the same. This quantity is essentially the Fourier transform of the symmetrized quadrature correlation function; this is the connection between the time and frequency domains referred to in the earlier report.

The present work here still seems to me to be very closely related to Ref 6, in particular. Both papers purport to calculate the symmetrized spectral density of an optical field quadrature coupled to a thermal oscillator.

I do not believe it is reasonable to expect a referee to reproduce lengthy calculations in detail; one can only judge from the clarity of the presentation and the plausibility of the conclusions. I find the present paper lacking in both respects. The issues with the introductory material have already been addressed.

I find the conclusion that observed sideband asymmetry being merely a coincidence of the simultaneous modification of the intrinsic damping with temperature to be unlikely. In order to make this claim more reasonable, experimental data should be used to specify how much this damping would have to vary with temperature, and assess if this is consistent with observed experimental variations in temperature-dependent damping.

Finally, given that Appendix C seems to claim that multiple published papers are in error, perhaps it would be more appropriate to write comments on these papers in order to elicit a response from the original authors of these papers. It seems more reasonable than expecting a referee to attempt to reproduce calculations across multiple apparently conflicting papers.

• validity: poor
• significance: poor
• originality: poor
• clarity: poor
• formatting: acceptable
• grammar: acceptable

### Report

Upon resubmission, the authors have made a number of changes to the main text. This includes shifting large sections of text to the supplementary information, and including a brief comment on experiments that leverage single photon detection. I think these changes improve the readability of the paper.

In the previous submission, referee 2 raised a number of criticisms of this work. I can certainly understand his/her perspective on certain aspects, in particular, that the subtle interpretation of sideband asymmetry has been the subject of numerous detailed papers over the last decade. For example, there is the PRA by by Kjetil Børkje (Ref 18) which details the interpretation of sideband asymmetry. There is also the review article by Clerk et. al. (Ref 14) which provides an extremely detailed account of quantum measurement theory. However, without going through the detailed calculations, I can't find any specific errors/contradictions in this manuscript that would negate its claims.

Quite generally, I think this subject is interesting and timely. As a result, I think it should be published in SciPost.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Strengths

In helping to make the clear the relationship between classical and quantum linear systems, the manuscript makes a valuable contribution to a topic of current interest.

None

### Report

I think the authors have done a good job of improving the manuscript based on the suggestions of the previous referees. I recommend the manuscript for publication.

### Requested changes

none

• validity: top
• significance: good
• originality: good
• clarity: good
• formatting: excellent
• grammar: good

## Comments

Matthew Davis on 2020-04-16
Category:
remark

From the editor This is a further comment delivered by email by the author of report 1 from the first round, and report 2 of the second round of refereeing. It is cut and paste as a comment by the editor with the author's permission. I have clarified that they are discussing the comments in report 2 from the first round, and report 3 from the second round of refereeing.

From the referee

I have read through the report submitted by the critical referee (Ref2). He/She has focused their criticism onto the interpretations made early in the paper (specifically of Eq. 1-2) and, in particular, where those interpretations sit relative to Ref. 14 and Ref. 6. I should point out that those two references dedicate (collectively) over 120 pages to the problem of measurement of quantum noise, with a significant fraction of the content being linked to observations of sideband asymmetry... So, to an expert in quantum measurement theory, perhaps this is a solved problem. Unfortunately, my expertise does not lie directly in this area, so I find it difficult to discriminate between these two interpretations.

However, Ref2 raised an important point that I had missed on my first review (Ref2's second-to-last point)! It relates to the temperature scaling of the observed sideband-asymmetry. In the "conventional" theory, the sideband-asymmetry is expected to go to zero at higher temperatures because of the difficulty in resolving 1 phonon in a bath of millions. However, on page 6 of the manuscript, in the paragraph before the conclusion, the authors state, "For real physical systems, the mechanical quality factor Q varies with the temperature, which leads to a temperature dependent asymmetry ... Whenever the quality factor decreases linearly with increasing temperature (at least for a given temperature range), then the qualitative temperature dependence of the asymmetry matches the standard result.". In this section, the authors claim that their calculations suggest that the temperature dependence observed in experiments arise purely from a linear change in the mechanical dissipation with temperature. I find this extremely unlikely. The most recent experimental observations of sideband-asymmetry were performed in a variety of materials (i.e. aluminium, silicon nitride, superfluid helium), each with a different scaling of mechanical dissipation with temperature. Furthermore, in contrast to the authors claim, all of those experiments have reported a temperature scaling that is consistent with the conventional theory.

In my opinion, these concluding remarks call into question the theory preceding it. As such, I'm now more aligned with the critical referee. The paper should not be published unless this issue is addressed. In this context, Ref2 makes a good suggestion; "... In order to make this claim more reasonable, experimental data should be used to specify how much this damping would have to vary with temperature, and assess if this is consistent with observed experimental variations in temperature-dependent damping."

Anonymous on 2020-04-20

Having read the latest reports from the other referees, I realize that I had not looked carefully enough at the manuscript in my first report. In particular, I was not aware that the issue of the classical nature of the measured spectrum for linear systems has already been treated in detail (especially reference 6).

For what the manuscript claims, the presentation is inadequate. As referee 2 states, the primary topic of the manuscript has already been treated in detail. Thus the novelty of the manuscript relies on the authors claim that multiple previous papers have significant errors, including calculations errors. This is presumably the reason that, in the introduction, they are so dismissive of previous work on the subject. The manuscript also contains another unusual claim, which is that the previous experimental observations of the expected temperature-dependent asymmetry is not due to the temperature dependence found in previous derivations, but due only to the temperature dependence of the mechanical damping rate. Both claims are sufficiently improbable that they require a detailed justification. This is why the presentation is wholly inadequate. As a previous referee states, if the authors believe that there are errors in previous papers sufficiently significant as to render the conclusions incorrect, one option is to write comment specifically on one of those papers, redoing the calculations correctly so that everyone can see exactly where the errors are. If the authors wish to write a single paper in which the claim significant errors in a number of papers they need to provide supplementary material, or appendices that gives the details of the calculations to make clear exactly what the errors are. Also, the claim about errors in previous works should be up front in the introduction, since it is a major part of the work; it is the claimed reason that previous work has come to a very different conclusion about the form of the output spectra.

The output spectrum obtained by the authors leads to the conclusion that the temperature dependence of the asymmetry has a completely different origin than previously thought. This also requires a much more detailed comparison with previous work, this time experimental. The claims of previous errors, and the claims of a new origin for the temperature dependence, as far as I can tell, go together. If one is incorrect so is the other, most likely. That’s why the authors should be checking their claims against previous work in much more detail.

So, as with the previous referees, I do not recommend publication without significant additions to the manuscript.

Matthew Davis on 2020-04-21
(in reply to Anonymous Comment on 2020-04-20)

As the editor, I am verifying that the comment above is by the author of report 1 from the second round of refereeing.