Adaptive-Step Perturb-and-Observe Algorithm for Multidimensional Phase Noise Stabilization in Fiber-Based Multi-Arm Mach–Zehnder Interferometers

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Review Report: algorithms-3286106
In this manuscript, the authors investigated phase noise control in a multi-arm Mach-Zehnder interferometer (MAMZI) system. They developed a feedback control protocol based on a modified perturb-and-observe (P&O) algorithm to monitor phase noise effects and counteract
the phase noise by optimizing the MAMZI’s output light intensity. The performance of this proposed algorithm was investigated for MAMZIs with 2 to 8 arms by simulations of the MAMZI devices with added phase noise.

This work presents a protocol for reducing noise in a MAMZI, addressing a significant challenge in this field. I have several questions regarding the feedback control algorithm, which I suggest the authors to clarify:

One of my concerns is about the major differences between the proposed algorithm and previous protocols, such as the one discussed in Ref. [22]. It appears that this work (Ref. [22]) also employs a phase compensation approach to maximize light intensity at the photon detectors. The authors may wish to comment on previous efforts and compare them with the current protocol to highlight advancements made with this new algorithm. Related to this, the
introduction section might benefit from a more comprehensive discussion of prior attempts to stabilize the phase in MAMZIs. Although there is some mention of hardware improvements, further background on previous stabilization approaches (especially on the algorithm side) could
enhance context.

In the P&O algorithm used in this work, the perturbation applied to phase control is dependent on noise strength (0.01 pi) [see Point (ii) in Sec. 4.2, p. 5]. I wonder whether this dependency is essential to the algorithm. If it is, the authors may want to explain how this information is extracted from the experimental setup. If not, they might consider rephrasing it to exclude this
detail from the perturbation algorithm description.

Furthermore, I suggest the authors expand on the construction of the feedback control protocol in the discussion around Fig. 2. For instance, clarifying how I_ref should be determined experimentally and explaining symbols (such as the +/- circuit notation) could improve readability (or combine it partially with the discussion of Sec. 4.2).

In addition, I also have a few minor comments:
1. In Sec. 3.1, the last sentence of the first paragraph, the statement, “Another essential variable is … can equalize it for all arms, allowing the constants tau_k to be considered as unitary values in (2)” [line 95], seems incorrect. I believe the authors meant to indicate that this adjustment makes tau_k to be independent of k, enabling it to be factored out, which makes the phase matrix to be a unitary matrix.
2. In line 132, “The input optical intensity (I^{in}I_0) is a constant”, I_0 seems to be redundant.
3. In Figs. 3 to 5, does the x-axis represent time? "AU" is unclear as a label and could be more informative.

Overall, this manuscript addressed a key challenge in MAMZI systems, stabilizing the phase across different paths in MAMZI. It proposed a feedback control protocol with promising results from numerical simulations. I recommend that the authors compare this protocol with previous approaches for phase noise stabilization in MAMZIs to clearly demonstrate the advancements of this work. I will consider recommending this manuscript for publication once these points are adequately addressed.

Comments for author File: Comments.pdf

Author Response

Thank you for your thoughtful feedback and for considering this manuscript for publication.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In this work, the authors address the challenge of phase noise stabilization in fiber-based multi-arm Mach-Zehnder interferometers (MAMZI) using an adaptive-step Perturb-and-Observe (P&O) algorithm. They examine the controllability of phase fluctuations, which are critical in applications such as telecommunications, spectroscopy, and quantum information, by employing a modified P&O algorithm to mitigate multidimensional phase noise.

They conduct numerical simulations up to 8x8 dimensions, demonstrating that their adaptive feedback approach can maintain high optical visibility and minimal steady-state errors, even in complex N x N interferometric systems. This study shows that MAMZI setups affected by multiparametric phase noise remain viable for high-dimensional quantum information protocols, achieving stability with minor adaptive modifications.

The conclusions of the manuscript confirm that the controllability of the MAMZI under phase noise is feasible for dimensions ranging from 2x2 to 8x8. They highlight that further refinement of the control model, including advanced algorithmic adjustments, could enhance stability indicators such as steady-state error and control response time, suggesting future work in this direction. 

The work is presented clearly with a logical progression from problem identification to solution demonstration and analysis including the realistic loss and polarization changes.

It offers valuable insights into phase noise control for MAMZI systems, with implications for high-dimensional quantum applications. Further exploration of algorithmic optimization would strengthen future work.

While the adaptive-step Perturb-and-Observe (P&O) algorithm is central to this study, the paper could benefit from a deeper exploration of why this specific algorithm was chosen and how it compares to alternative approaches for phase noise stabilization.

I would also suggest the authors to consider adding a brief discussion on the limitations of the P&O algorithm, especially regarding scalability or response to rapidly varying phase noise, and compare it with other control strategies.

What can be said about potential discrepancies between simulation and real-world performance?

The perturb&observe algorithm reminds me the (quantum) Zeno dynamics, especially the “rotate-measure” iterations for driving a quantum system to a target state. What can be said about a potential analogy of the present approach to Zeno dynamics?

I also checked the iThenticate report in detail. The 9% is due to some template phrases of the journal, and some very common phrases/words in the text, so there is no issue at all.

Finally, I would suggest only a quick and minor revision before publication.

Author Response

Thank you for your thoughtful feedback and for considering this manuscript for publication.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

In the revised manuscript, the authors addressed the concerns that I raised in my previous review report. Specifically, the authors presented a comparison between the proposed algorithm with the previous schemes. They highlighted the major limitations of the previous schemes and clearly discussed the improvement of the current proposed scheme. Overall, the proposed scheme discussed in this manuscript shows an improved control scheme to stabilize the phase noise in a multi-arm Mach-Zehnder interferometer system. The proposed scheme is numerically verified. Therefore, I suggest accepting this manuscript.