Peer review of: Two-dimensional optomechanical crystal cavity with high quantum cooperativity (2020)

Reviewers' comments: 

Reviewer #1 (Remarks to the Author): 

This paper provides a clear description of the design, fabrication and measurement of two-
dimensional optimechanical crystals that are cooled close to their mechanical quantum ground 
state. Measurements are reported for the optical and mechanical loss, mechanical heating due to 
the optical probe signal, and estimates made for the quantum cooperativity for optimechanical 
state transfer. 

This paper reports an astounding mechanical quality factor in excess of one billion at 10 GHz, 
reproducing, and closely parallel to, an earlier manuscript from the same group (ref 32), which 
dealt with low temperature behavior in one-dimensional optimechanical crystals. This earlier work, 
with even higher reported mechanical Q's at similar frequencies, apparently has not yet been 
published, which is unfortunate, as this earlier paper represents a signature result with a detailed 
analysis very similar to this work. The complicated thermal modeling and behavior reported here is 
quite similar to that earlier work. 

In the submitted paper here a similar level of mechanical quality factor is achieved at the lowest 
optical excitation powers as in Ref 32, with this measurement similarly confounded by heating due 
to the laser probe. However in this two-dimensional design the heating is reduced sufficiently that 
more promising performance is predicted for somewhat more optimized devices. The paper is 
mostly very clear and shows a number of new results; the supplementary also provides a wealth of 
experimental detail that is quite welcome. I would support publication in Nature Communications 
with some mandatory changes as below. 

Please provide an explicit expression for the cooperativity C when it is introduced on p 1 

Please provide an expression and reference for the extraction of the optimechanical coupling rate g 
from the linewidth dependence on cavity photon number. 

Many parameters are calculated from the data without any consideration for the uncertainties in 
the arrived-at results; just as an example, the mechanical decay rates given on p 6 are given with 
4 significant figures; I find it hard to believe, given the data, that these can be determined to 1 
part in 10^4. Please provide uncertainties for all such derived quantities. 


Reviewer #2 (Remarks to the Author): 

The authors demonstrate a two-dimensional (2D) Si optomechanical crystal (OMC) cavity with a 
high quantum cooperativity at millikelvin temperatures. The structure is well band-engineered in 
both photonic and phononic systems and achieves high optical and mechanical quality factors. This 
group investigated similar topic in a 1D nanobeam OMC structure before. In this study, design and 
fabrication of a 2D OMC cavity with a high acoustic cavity Q and much better thermal conductance 
with the cold bath reservoir result in a higher quantum cooperativity, which is the most relevant 
figure-of-merit for quantum optomechanical applications. The progress from the 1D to 2D OMC 
cavity is clear as stated in the last paragraph in p. 1, and this work achieves an effective quantum 
cooperativity of unity. The level of experiments and analyses are high, and the discussion is 
reasonable and convincing. Therefore, I recommend publication after minor revision. 

1. Regarding the quantum cooperativity, the relevant threshold for coherent photon-phonon 
interaction is unity. How high quantum cooperativity is necessary for realistic use in applications? 
Please pick up one example of application and discuss how far/close current technology level is. 

2. In Fig. 4b, np is proportional to nc^0.3. What determines this power-law of 0.3? Is it reasonable 
that 1D and 2D have the same power-law? 

3. (minor) In Fig. 1c, the broken lines are too thin and could not find them when I printed. I 
recommend making it thicker. 



Reviewer #3 (Remarks to the Author): 

I have read the manuscript « Two-Dimensional Optomechanical Crystal Cavity with High Quantum 
Cooperativity” from Prof Painter and co-workers. Briefly, the manuscript describes a new type of 
two-dimensional optomechanical crystal cavity engineered for improving the low temperature 
thermalization properties and thereby maximize the optomechanical cooperativity (which 
represents the weight of quantum fluctuations respective to that of classical ones). This work 
reports on the background, design and fabrication of the device, as well as on the optomechanical 
and thermal experimental characterizations and calibration. The context and hypothesis of the 
work are remarkably clear, as well as the adopted scientific methodology. The quality of the 
experimental results and their agreement with theoretical modelling are high. This manuscript 
represents an important piece of work to the field and beyond, thanks to the general problematic 
being addressed (heat bath engineering applied to the design of quantum coherent devices) and 
its very accessible presentation. In light of the above remarks, I strongly recommend publication 
of this major piece of work in Nature Communications. 
The manuscript is organized into 6 parts: The authors thoroughly introduce the context of their 
work in the first part. Details on the design methodology and fabrication are given in the second 
part, along with the presentation of the experimental setup. The optomechanical characterization 
and calibration is provided in the third part. The fourth part presents the measurement and 
analysis of the thermal properties of the optomechanical system under continuous optical driving. 
The fifth part reports the measurement of the phonon occupation and corresponding effective 
quantum cooperativity under continuous optical drive. The authors conclude in the last, discussion 
part. 
Introduction 
In the first, introductory part, the authors set their work into the general context of (quantum) 
optomechanical systems. They clearly describe the importance of developing high cooperativity 
optomechanical systems operating at ultra-low temperatures, notably in the perspective of 
building quantum hybrid interfaces between microwave frequency logic circuits and optical 
quantum communication channels. The intrinsic problematic of 1-dimension optomechanical 
crystal cavity systems, showing very high optomechanical coupling rates but reduced thermal 
conductivity (and which therefore greatly suffer from the absorption-induced heating) is very well 
introduced. 
Comment: I have no specific comment on this very well written part. 
Design/fabrication 
In the second part, the authors present the concept of their new device aiming at preserving a 
very high optomechanical coupling while simultaneously increasing the thermal conductivity, 
enabling to load significantly more intra-cavity photons, and therefore resulting in a much 
increased effective optomechanical cooperativity. Their strategy relies on making use of the 
frequency-dependent density of phonon states within a 2D phononic bandgap structure consisting 
of a 2D optomechanical crystal cavity. The fabrication steps, the analysis of the experimentally 
measured geometrical properties and the simulated optomechanical parameters are clearly 
presented in that section, confirming the relevance of the proposed approach as far as the 
optomechanical properties are concerned. 
Comment: I have a minor comment regarding Fig 2.c: I would advise the authors to mind the use 
of colours (and to maybe compliment it with different point styles) for the colours-blind readership. 
Optomechanical characterization 
In this third part, the authors present the optomechanical characterization of the newly fabricated 
devices. Optomechanical characterization is reported both at room and cryogenic temperature, 
including the optical Q-factor, mechanical resonance frequency, optomechanical coupling rate and 
mechanical Q-factor. In particular, the authors pay a great deal of attention for avoiding dynamical 
backaction effects by performing ringdown measurements, which besides showing decreased 
sensitivity towards dephasing, enables to be “in the dark”, thereby suppressing the contribution of 
any delayed dynamical backaction. The authors notably report a massive mechanical Q-factor 
exceeding 1 billion, with a mechanical resonance frequency in the 10 GHz range. 
Comment: I have a comment on this part: The authors report measurements relying on non-linear 
optomechanical amplification of a resonant phase modulation, which is sometimes also referred to 
as “OMIT” measurement. I believe such experiment to be nontrivial to a broad readership and 
would suggest a more pedagogic presentation, besides the (rightfully) given references. Along 
these lines, I would recommend a description of the solid line fits featured on Fig. 3(b). 

Thermal measurements 
In this fourth part, the authors present a study of the thermal properties (both effective damping 
rate and phonon occupation) of the OMC cavity as a function of the input optical power. The 
authors essentially identify two regimes for the sensitivity of the mechanical properties as a 
function of the intracavity photon number. Importantly, they establish a connection between the 
effective temperature (number of phonons) and the mechanical damping rate, thereby describing 
the effect of the absorption as that of an effective “hot bath”. 
Comments: I have a few comments on this part. 
a) I find the last sentence of the second paragraph (right column) on page 5 somehow too long 
and not easy to understand. I would recommend the authors to try to simplify this sentence. On 
the power low: could the authors maybe comment on a more fundamental solid-state physics 
point of view maybe? 
b) 3rd paragraph right column on page 5: the authors refer to Fig. 3(b) instead of Fig. 4(b). 
c) page 6 right column: The authors state “at the lowest power (…) the linewidth saturates to a 
constant value; this is not entirely clear to me that the data confirm this. 
d) On \gamma_\phi: how do the authors prove this to be dephasing (besides being much larger 
than the “zero power” ring down value)? Did the authors maybe perform ringdown measurements 
(e.g. by means of a pump-probe configuration)? Could the authors better explain why the two 
regimes of temperature are fitted using different models? Where do the authors set the “cut off” 
between these two regimes? 
e) End of 2nd paragraph right column page 6: The last sentence sounds somehow cryptic. 
Measurement of the optomechanical cooperativity 
In the fifth part, the authors report on optomechanical thermometry measurements, and 
corresponding optomechanical cooperativity. Their notably use their analysis to attribute the 
(significant) excess of thermal occupation to the phonon population within the coupling waveguide, 
which contaminates that of the cavity under the effect of optical absorption. The (somehow 
empiric) model of the authors shows very satisfying agreement, which certainly paves the way not 
only to further technological improvement but for deeper understanding and assessment of the 
spatial location of decoherence in ultra-sensitive optomechanical systems. 
Comments: I have a comment/question on this part: Page 7 left column 3rd paragraph : did the 
authors tried to optimize the sideband ratio to see if this could be beneficial to the cooperativity? 
Discussion 
In the last part, the authors briefly discuss their result and put them into perspective with future 
possible research pathways that could benefit from them. This part is convincing and very well 
written. 
Comment: I have one minor comment: An 68 fold increase of the thermal conductivity is 
mentioned, which disagrees with the number (42) given in the main text body.