Single-Port Homodyne Detection in a Squeezed-State Interferometry with Optimal Data Processing

Round 1

Reviewer 1 Report

I recommend in favor of publication of this manuscript, provided a number of improvements are made on it.

 

I sustain publication of the paper in spite of the fact that its content is little more than an exercise, because it deals with a problem which is of great practical interest, though perhaps of secondary fundamental importance:

 

This is the question: when homodyne detection is performed at one out-port of an interferometer where one feeds at the input a mix of coherent light with some small amount of squeezed vacuum (a procedure that Caves showed to give the best sensitivity), and then runs an appropriate data processing (as proposed by Schäfermeier et al.: divide the measurement into a number of bins and then construct the inversion estimator associated to one of the outcomes), one can simultaneously overcome the known limits to sensitivity and resolution.

 

This methodology is well known and routinely applied by quantum optics experimentalists to get super-sensitive phase measurements. What is typically neglected, however, is that in practice both sensitivity and resolution depend crucially on the size of the beans. The authors derive, for the case of three bins, the optimal size, resorting to a specific form of maximum likelihood estimator.  The results are reliable, as on the one hand they confirm the numerical outcome of simulations, and on the other they also prove to be asymptotically optimal. I expect them to be of very positive use to improve the quality of experiments, and the method to lend itself to generalizations and improvements.

 

As an overall comment, I say that the authors should be a little more critical about the whole manuscript: for example, can one do better than choosing Fisher’s entropy (Cramér-Rao indication) in the estimate of the phase sensitivity? what kind of effort would increasing the number of beans imply? is it conceivable to have a better approximation of the Maximum Likelihood Estimator than Zhou, L. K et al.’s method?  Above all, however, the paper requires a thorough revision of English spelling, grammar and syntax.

 

Author Response

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Reviewer 2 Report

This paper reports improved sensitivity and resolution of phase measurement by interferometry using 3 bins for processing the measurement outcomes. This concept was proposed by Ref[10] in 2018 with binary measurement outcomes and the detailed scheme for obtaining the optimal results on demand has been demonstrated. The authors took the same approach and considered the case of 3 bins. 

I do not think this work contains sufficient new findings to publish. This is reflected in their conclusions, in which the authors claimed that 
"As a three-outcome measurement, this kind of data processing can improve the phase resolution and the phase sensitivity beyond their associated classical limits." Actually this has been pointed out by using a two-bin scheme already.
Also, "Our analytical results show good agreement with the exact numerical results as long as the ratio ..." I am not impressed at all by the achievement of the analytical solutions without any physical insight. As a theoretical work, this paper does not present any new findings worthy to note. 

In addition, the readability of this paper is very low due to the poor presentation. The writings contain too much jargons that are not easy to understand for readers not in this field. The main text are mostly mathematical derivations without physical meanings of used variables and formulae. Moreover, the motivations and assumptions in these calculations are not very clearly stated. Though the authors provided some references for each equation and calculated data, it is authors' responsibility to provide self-contained background information. If the authors do feel that some derivations are not suitable to be included in the main text, they should provide them in the appendices. 

In summary, I judge that this paper should be rejected for publication.

Here I also list some other points the authors should consider to improve:

-There are many symbols used without clear definition, such as \theta and \lambda in page 1, k and a in page 2 and many others when they are mentioned for the first time.

-In Eq. (3), there is no mentioning what are \mu and \nu. Are they parameters of the squeezed state?

-|\phi_{out}\rangle = U(\theta)|\phi_{in}\rangle. There is no definition of U(\theta) 

-It is mentioned that Eqs. (3-6) hold for arbitrary Gaussian state but is difficult to see from the assumption and derivation.

-There is no mentioning about what is \Pi. It is stated that "\mu_k denotes the eigenvalue of k-th outcome.", but the eigenvalues of what? Why the eigenvalues depend on bin size a?

-What is \mu_{\oslash}?

-There is the sensitivity for 3-bin case in Fig. 1(d) but no resolution data is shown in Fig. 1(c) 

-Caption in Fig. 1(d): "Horizontal lines in (d) the the SNL and the analytic result predicted by Eq. (21)."  But which is which?

-"As shown by the red dotted line of Fig. 1 (d), one can find that the divergence of the sensitivity can be removed using the CRB of the three-outcome measurement..., where the CFI F(q) has been defined by Eq. (2)..."
The authors only gave the explicit results for binary outcome case but not 3-outcome one. This statement is not evident. 

-"The above result is given by substituting Eq. (10) into Eq. (2) and calculating the CFI at \theta = 0."
It is difficult to see how \theta = 0 yields the maximal value of F_{max} and the result can be obtained from Eqs. (10) and (11). 

-"Numerically, the best sensitivity...is obtained for different value of a ..., which may occur at \theta_{min}\neq 0." In what situations, the minimal F value occurs at \theta=0?

-Regarding the resolution and sensitivity: It would be clear to see a better resolution, which controlled by "a", would result in a poor sensitivity and vice versa as shown in the 2-bin case. However, this is not true for the 3-bin case. Instead, the best resolution occurs at a rather small "a=0.18", which gives a good resolution as well. The authors could try to give some explanations on this. 

-Regarding the test of maximum likelihood estimation, how the preparation and sampling for the noisy signals is not clearly stated. For example, how large is the variation in random numbers? From the data presented in Fig. 4, I see that N=50 the simulated result is close to CRB. It seems to me that it should related to the variation of random numbers you used.    

Author Response

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Reviewer 3 Report

The authors discussed homodyne detection for interferometric-phase estimation. They considered the situation of three-bin/outcome and obtained an analytical expression for the optimal bin size by minimizing the Cramer-Rao bound. They further showed that the maximum-likelihood phase estimator is asymptotically efficient.

While the calculations are technically correct, I feel that the content
of this article merely serves as a follow-up exercise of known three-outcome 
homodyne measurement schemes. Further, the finding that the maximum-likelihood phase estimator is asymptotically efficient is not surprising at all as it is simply a sanity check of well-known results in statistics.

Nevertheless, since the article has no obvious flaws, I suppose it can be 
accepted by MDPI Photonics as a technical article.

Author Response

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Reviewer 4 Report

In this paper, the authors investigated high-precision homodyne measurement with a single output port. In particular, the authors analyzed the optimal value of the bine size in the divided three bins. Even though such a study on the bin size, or in general the data processing over the measurement outcome,  has been known for the controls on the phase resolution, see the cited Refs. [10, 11]. This paper examined the bin size for the three outcomes in details, including the corresponding Fisher information, error-propagation formula, and the average photon numbers. The comparison with approximated MLE is also provided, to show the validation of their results. 

 

However, I am curious about the main result on the best sensitivity obtained when  the average number of the squeezed vacuum is small than $0.55 \bar{n}, \sqrt{\rho}$. Here, the purity of squeezed vacuum is introduced in Line 39th, but without introduction of any non-squeezed noises. In contrast to the mixed states, it is too ideal to have an input squeezed vacuum as the pure states. What would be the resulting sensitivity when a mixed input state is considered?

 

In short, the paper is well written, with the analyses on the proposed scheme in details. If the issues listed above can be addressed, I will be happy to give my recommendation.

 

 

 

 

Author Response

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Round 2

Reviewer 2 Report

The revision looks good. I recommend accept in the current form.