Directional Elastic Pseudospin and Nonseparability of Directional and Spatial Degrees of Freedom in Parallel Arrays of Coupled Waveguides

Round 1

Reviewer 1 Report

In this manuscript the authors report an experimental study of the modes associated with dynamical propagation of elastic waves along an array of coupled metallic waveguides (WGs). As waves can propagate in both directions of the WGs, the propagation direction can be described as a two level degree of freedom (like a quantum spin 1/2), and represented through the formalism of pseudo-spins. Using the machinery of quantum physics, an elastic wave in this system can be expressed in terms of a state in a Hilbert space that is the tensor product of two subspaces: one of dimension N, with N being the number of WGs, and another of dimension 2, associated with the pseudospin degree of freedom. As shown theoretically in Ref. [42] by some of the authors, modes of these systems are separable in terms of their spatial part and their “spin part”. In addition, superpositions of these modes can be engineered, leading to excitations that are not separable anymore. Of particular interest are those superpositions that have the same expression of Bell states.

I have to admit that I find a large overlap between the content of this paper and Ref. [44] by the same group of authors. In the latter, they report a very similar study, on the same experimental system. Some figures are the same, with text in the caption being very similar (if not identical). See for instance Fig. 3 in the present manuscript and Supplementary Fig. 1 in [44], or Fig. 5 and Fig. 2 in [44]. I did not appreciate that in this manuscript there is discussion about the relationship between the two studies. What is the novelty if this article, if compared to Ref. [44]? The latter is cited only jointly with Refs. [42,43]. A Referee should be helped in understanding how a paper builds on past results, in particular when the similarities are so important.

Let me add a disappointing detail. By reading Referee’s reports associated with Ref. [44], published by the journal and available at this link.

I was impressed to read the following remark by Referee 1

<< In the discussion the authors make an inaccurate statement: “for the case of laser light the coupling between the degrees of freedom is weak and, hence, realizing all the possible relationships for the nonseparable superpositions is not possible.” Q-plates are a quite flexible and efficient tool for preparation of non-separable spin-OAM states of a laser beam thanks to the strong spin-orbit coupling achieved in liquid crystals. There are also the so-called S-plates for preparation of radially polarized beams, that are also non-separable spin-OAM states. With these devices complemented by birefringent wave plates, any spin-OAM state can be easily realized. I suggest the authors remove this kind of assessment in the manuscript. >>

Whose response was 

<< Response: The above mentioned statement has now been removed in the revised manuscript.>>

I was formulating the same remark after reading the following sentence in the present manuscript:

“In particular, in the field of optics, degrees of freedom of photon states that span different Hilbert spaces can be made to interact in a way that leads to local correlations [5,7–17]. These systems have a drawback in the sense that the coupling between the degrees of freedom is weak and, hence, realizing all the possible relationships for the nonseparable superpositions, called Bell states, is not possible.”

Hence, the last sentence was (in a very similar formulation) already contained in the submitted version of Ref. [44], yet removed after the Referee’s remark. Why do the authors keep it in the present manuscript, even though they agreed that is was not accurate and removed it from the revised version of Ref. [44]?

The large overlap between the two manuscript and the lack of any discussion between their connections (a joint citation with other studies is not sufficient) makes me quite skeptical about suitability of this manuscript for publication by any scientific journal. 

Let me add something about the content of the paper. I disagree with the choice of referring the spatial modes of this system as Orbital Angular Momentum  (OAM) modes. In optics, these spatial modes have a very specific distribution of the optical field in the plane perpendicular to the propagation direction. Specifically, these are characterised by a phase factor $e^{i m \phi}$, with m being an integer and $\phi$ the azimuthal angle in the transverse plane. As such, these modes feature a twisted wavefront. This discrete set of phase distributions can be associated with periodic variables only, such as an angle. In the system studied in this manuscript, the spatial coordinate is the index identifying the WG. In Ref. [42] WGs were disposed on a circle, so that the first was coupled to the last. In that case, I agree that there is an analogy between the spatial modes of the system and optical modes carrying OAM. In the present article, as in Ref. [44], the three WG are aligned in a linear setting and there is no coupling between the first and the third, as these are not neighbour. Optical counterparts of this system can be found in a variety of photonic setup, the authors can refer for instance to the very large number of experimental and theoretical papers by Prof. Longhi (scholar page) or by Prof. Szemeit (scholar page). Spatial modes of these systems are simple plane waves in the plane spanned by the directions along and across the WGs.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

"Directional Elastic Pseudospin and Nonseparability of 2 Directional and Orbital Angular Momentum Degrees of 3 Freedom in Parallel Arrays of Coupled Waveguides"
by Hasan et al.

The paper proposes a different approach to the interpretation of elastic eigenmodes of macroscopic mechanical structures. A formalism that transforms the wave equation of the acoustic waves into the form of a formal square root of the wave equation that resembles the Dirac equation is developed. The concept of pseudospin is introduced: modes propagating in the opposite direction are treated as two separate degrees of freedom. The Dirac-like equation is analysed and solutions are discussed. Last, the authors show experimental realization in which two degenerate elastic eignemodes are excited in a superposition.

The results of the paper seem to be correct and the analysis potentially brings an interesting insight (although the results seem to be fully understandable using standard approaches and are not too surprising). However, some critical clarification is needed before the paper can be published, as detaild below.

Comments:

It is not clear to me if the authors think about some specific use of the "classical entanglement" (as opposed to the quantum entanglement where many particles can be entangled and yield non-classical correlations) or if the study only aims at mathematical aesthetics. The authors should clearly motivate their work (and why the standard eigenmode analysis is not suitable).

Overall, the paper studies classical eigenmodes of a macroscopic oscillating system. The authors themselves provide numerical solutions of a classical discretised equation of motion (a discrete form of a wave equation in Eqs. 10 and 11 of the manuscript) for the system consisting of several coupled rods. To my understanding, the analysis utilising the Dirac equation is not necessary for understanding the eigenmode structure: The resulting superposition of states shown by the authors can be understood as a consequence of the degeneracy of the eigenmodes belonging to different bands of the dispersion relation of the system (any superposition of degenerate eigenmodes of a linear operator is a valid eigenmode of the system - the superposition can be selected by appropriately exciting the system, as the authors experimentally demonstrate).

The authors should clarify the exact mathematical connection between the vector Psi_{2Nx1} and u_{Nx1} (i.e., please, state the exact mathematical formula connecting the two). Both must represent the same classical quantity, but the former is derived from a different equation. What is the exact physical meaning of Psi_{2Nx1}? Also, the symbol +- appears in Eq. 2 and equations/solutions derived from here. The sign, however, never seems to be reflected in the definition of vectors Psi_{2Nx1} (or s_{2x1}^(n)). This point should be clarified and precise mathematical definition of which sign should be used should be clearly stated. This point is critically important as the analysis of the Dirac equation is the main focus of the paper.

I would like to point out that the "direction of propagation" cannot be used in the finite system as a label for eigenmodes. This naturally leads to the need to introduce counter-propagating waves as eigensolutions (Eq. 6) - in this case the degeneracies of eigenfrequencis are not guaranteed. How do the authors ensure that different phonon bands are perfectly degenerate for different k_1 and k_2 (as needed to excite the superposition state)? Is it just a coincidence (shown in Fig. 5) or is it possible to reach such degeneracies on demand (e.g. by straining the system...)?

"..a vector which components..." is incorrect (use: "whose" or "components of which").

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

This revised version mitigates the criticisms that were raised in my first report. It is clear now the link between this manuscript and the rapid communication that anticipated its most relevant results. It would have been great having this clear since the first version.

Author Response

We thank the reviewer.

Reviewer 2 Report

The manuscript has been improved by the authors, a clear motivation has been added (although, for example, collapse upon measurement is largely a feature exploited in quantum mechanics - e.g. in quantum teleportation - not a problem), however, I feel that some of my comments have been misunderstood.

For example:

?2×1(?)=?0(√??+??±√??−??) strictly speaking implies that ?2×1(?) must be simultaneously equal for both + and - sign (which it is not, I believe). This is a formal detail that can be resolved by adding an index as follows:

?2×1(?)(±)=?0(√??+??±√??−??). 

I believe that the authors should use more rigorous mathematical notation (mainly due to the mathematical character of the paper). Please, implement the changes. 

Similarly, 

an explicit expression for Psi_{2Nx1}(u_{Nx1}) has not been added. A single equation would be sufficient (not an explanatory paragraph). As an example (the example is likely wrong and only serves an illustrative purpose- please use the correct version), something along the following lines should be added:

Psi_{2Nx1}(k)=(?2×1(|k|, 1)u_{Nx1}(|k|); ?2×1(|k|, 2) u_{Nx1}(-k)).

Could a similar (but corrected) EXPLICIT equation (including all the appropriate constants and coefficients) be added to the manuscript to increase the clarity? This would be a large improvement upon the latest version of the text. I consider this an point important. 

After the above changes are implemented, I believe the manuscript can be published. 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf