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arxiv: 2604.06676 · v1 · submitted 2026-04-08 · ⚛️ physics.optics · quant-ph

Steady-State Statistical Modeling of Digitally Stabilized Laser Frequency with Markov-State Feedback

Swarnav Banik , Elliot Greenwald , Xing Pan This is my paper

Pith reviewed 2026-05-10 18:16 UTC · model grok-4.3

classification ⚛️ physics.optics quant-ph
keywords Markov chainlaser frequency stabilizationdigital feedbacksteady-state distributionfrequency noisequantized actuatorphotonic integrated circuits
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The pith

A Markov-state model computes steady-state actuator and frequency distributions directly from the unit eigenvalue of the transition matrix in digital laser locks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper replaces continuous-time control theory with a discrete-time Markov chain whose states are the quantized actuator values and whose transitions depend on the discriminator curve, noise statistics, and digital control rules. The steady-state probability distribution over these states is the unit-eigenvalue eigenvector of the transition matrix, so all stability metrics follow immediately without running long stochastic simulations. The formulation is exact for white frequency noise sampled at decorrelated times; correlated sampling merely inflates actuator variance while leaving the mean unchanged. Colored noise produces sampling-dependent shifts in both mean and variance, marking the boundary of the memoryless approximation.

Core claim

The steady-state actuator and locked-laser frequency distributions are obtained directly from the unit-eigenvalue solution of the transition matrix, providing immediate access to key stability metrics without long time-domain simulations.

What carries the argument

The discrete-time Markov chain whose states are quantized actuator levels and whose transition probabilities are set by the frequency discriminator response, noise statistics, and implemented control logic.

Load-bearing premise

The laser system can be treated as a memoryless Markov chain whose transition probabilities are completely determined by the discriminator, noise, and control logic.

What would settle it

Compute the predicted steady-state actuator histogram from the unit-eigenvalue eigenvector and compare it to the histogram measured in a long experimental time series or Monte-Carlo simulation under decorrelated white-noise sampling; systematic mismatch falsifies exactness.

Figures

Figures reproduced from arXiv: 2604.06676 by Elliot Greenwald, Swarnav Banik, Xing Pan.

Figure 1
Figure 1. Figure 1: Schematic of the digital laser frequency stabilization loop. The free [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Description of the Markov method and comparison with time-domain [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of Markov and time-domain simulation variances for the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Deviation between Markov-predicted and time-domain steady-state [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Unified Voigt-based model for photonic frequency discriminators. (a) [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

Laser frequency stabilization is conventionally analyzed using continuous-time control theory, which accurately models analog feedback but is insufficient for digital implementations where quantization, sampling, and stochastic noise shape the dynamics. In modern digital laser systems, such as Photonic Integrated Circuit (PIC)-based lasers, finite discriminator and actuator resolution, sampling delays, and measurement noise introduce stochastic behavior that deterministic models do not capture. We present a discrete-time Markov-state framework that models the evolution of the quantized actuator in a digital laser frequency lock, with state-transition probabilities determined by the frequency discriminator response, noise statistics, and implemented digital control logic. The steady-state actuator and locked-laser frequency distributions are obtained directly from the unit-eigenvalue solution of the transition matrix, providing immediate access to key stability metrics without long time-domain simulations. For white frequency noise, we show that the Markov formulation is exact under decorrelated sampling and update schemes, while correlated discriminator sampling introduces a predictable inflation of actuator variance without shifting the operating point. In the presence of colored noise, long-range temporal correlations induce sampling-dependent deviations in both actuator mean and variance, defining the regime of validity of the memoryless Markov description. This framework provides a compact and physically transparent tool for analyzing and optimizing digitally stabilized lasers in integrated photonic systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces a discrete-time Markov-state framework for modeling the quantized actuator dynamics in digitally stabilized laser frequency locks. Transition probabilities are constructed from the frequency discriminator response curve, noise statistics, and the implemented digital control law. Steady-state actuator and locked-laser frequency distributions are extracted directly as the unit-eigenvalue eigenvector of the resulting transition matrix, yielding stability metrics (means, variances) without requiring long time-domain simulations. The model is stated to be exact for white frequency noise under decorrelated sampling and update schemes, while correlated sampling produces predictable actuator-variance inflation without mean shift; colored noise induces sampling-dependent deviations in both mean and variance, delineating the memoryless Markov regime.

Significance. If the central derivations hold, the work supplies a compact, physically transparent analytical tool for optimizing digital frequency stabilization in photonic integrated circuits, replacing computationally intensive Monte-Carlo runs with a single linear-algebra step. The explicit statement of the exactness conditions for white noise and the quantitative prediction of deviations under colored noise constitute a clear advance over purely deterministic control-theory treatments.

minor comments (2)
  1. A brief appendix or supplementary note showing the explicit construction of the transition matrix for a low-dimensional example (e.g., 3–5 actuator states) would help readers verify the mapping from discriminator curve and noise PDF to matrix entries.
  2. The manuscript would benefit from a short comparison table contrasting the Markov-derived variances with those obtained from direct time-domain integration for both white and colored noise cases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the Markov-state framework, and recommendation to accept. No major comments were raised that require specific responses or revisions.

Circularity Check

0 steps flagged

No significant circularity; standard Markov analysis applied to defined model

full rationale

The paper defines a discrete-time Markov chain whose transition probabilities are set by the frequency discriminator response, noise statistics, and digital control logic. The steady-state actuator and frequency distributions are extracted as the unit-eigenvalue eigenvector of this matrix, which is the standard linear-algebra result for any Markov chain and does not reduce to a fitted parameter or self-referential definition. The manuscript explicitly states the regime (white noise with decorrelated sampling) where the memoryless property holds exactly and notes predictable deviations for colored noise, without invoking self-citations or prior author theorems as load-bearing premises. No step renames a known empirical pattern, smuggles an ansatz via citation, or presents a prediction that is statistically forced by construction from the inputs. The derivation is therefore self-contained once the transition matrix is accepted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the assumption that laser frequency dynamics under digital control admit a memoryless Markov description; transition probabilities are taken as given from discriminator, noise, and logic without further derivation shown.

axioms (2)
  • domain assumption Laser actuator evolution under digital feedback and noise can be modeled as a discrete-time Markov chain with states corresponding to quantized actuator levels.
    This is the core modeling choice enabling the transition matrix and unit-eigenvalue solution.
  • domain assumption For white frequency noise, decorrelated sampling and update schemes make the Markov description exact.
    Stated as the condition under which the model holds without approximation.

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Works this paper leans on

26 extracted references · 26 canonical work pages

  1. [1]

    Laser phase and frequency stabilization using an optical resonator,

    R. W. P. Drever, J. L. Hall, F. V . Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,”Applied Physics B, vol. 31, no. 2, pp. 97–105, Jun 1983. [Online]. Available: https://doi.org/10.1007/BF00702605

    work page doi:10.1007/bf00702605 1983

  2. [2]

    An introduction to Pound–Drever–Hall laser frequency stabilization

    E. D. Black, “An introduction to pound–drever–hall laser frequency stabilization,”American Journal of Physics, vol. 69, no. 1, pp. 79–87, 01 2001. [Online]. Available: https://doi.org/10.1119/1.1286663

    work page doi:10.1119/1.1286663 2001

  3. [3]

    High-coherence parallelization in integrated photonics,

    X. Zhang, Z. Zhou, Y . Guo, M. Zhuang, W. Jin, B. Shen, Y . Chen, J. Huang, Z. Tao, M. Jin, R. Chen, Z. Ge, Z. Fang, N. Zhang, Y . Liu, P. Cai, W. Hu, H. Shu, D. Pan, J. E. Bowers, X. Wang, and L. Chang, “High-coherence parallelization in integrated photonics,” Nature Communications, vol. 15, no. 1, p. 7892, Sep 2024. [Online]. Available: https://doi.org/...

    work page doi:10.1038/s41467-024-52269-7 2024

  4. [4]

    Modulation-free laser stabilization technique using integrated cavity-coupled mach-zehnder interferometer,

    M. H. Idjadi, K. Kim, and N. K. Fontaine, “Modulation-free laser stabilization technique using integrated cavity-coupled mach-zehnder interferometer,”Nature Communications, vol. 15, no. 1, p. 1922, Mar

    work page 1922

  5. [5]

    Available: https://doi.org/10.1038/s41467-024-46319-3

    [Online]. Available: https://doi.org/10.1038/s41467-024-46319-3

    work page doi:10.1038/s41467-024-46319-3

  6. [6]

    Quantised output-feedback design for networked control systems using semi-markov model approach,

    W. Sun and Z. Ning, “Quantised output-feedback design for networked control systems using semi-markov model approach,”International Journal of Systems Science, vol. 51, no. 9, pp. 1637–1652, 2020. [Online]. Available: https://doi.org/10.1080/00207721.2020.1772400

    work page doi:10.1080/00207721.2020.1772400 2020

  7. [7]

    Dynamic output feedback quantization control of a networked control system with dual-channel data packet loss,

    F. Zhang, M. Hua, and M. Gao, “Dynamic output feedback quantization control of a networked control system with dual-channel data packet loss,”Mathematics, vol. 11, no. 11, 2023. [Online]. Available: https://www.mdpi.com/2227-7390/11/11/2544

    work page 2023

  8. [8]

    Observer-based quantized sliding modeH ∞ control of markov jump systems,

    M. Shen, H. Zhang, and J. H. Park, “Observer-based quantized sliding modeH ∞ control of markov jump systems,”Nonlinear Dynamics, vol. 92, no. 2, pp. 415–427, Apr 2018. [Online]. Available: https://doi.org/10.1007/s11071-018-4064-x

    work page doi:10.1007/s11071-018-4064-x 2018

  9. [9]

    Design of feedback control for networked finite-distributed delays systems with quantization and packet dropout compensation,

    W.-C. Hsu, L.-W. Lee, K.-H. Tseng, C.-Y . Lu, C.-W. Liao, and H.-N. Shou, “Design of feedback control for networked finite-distributed delays systems with quantization and packet dropout compensation,” Discrete Dynamics in Nature and Society, vol. 2015, no. 1, p. 158972, 2015. [Online]. Available: https://onlinelibrary.wiley.com/doi/ abs/10.1155/2015/158972

    work page doi:10.1155/2015/158972 2015

  10. [10]

    Feller,An Introduction to Probability Theory and Its Applications, V olume 1, 3rd ed., ser

    W. Feller,An Introduction to Probability Theory and Its Applications, V olume 1, 3rd ed., ser. Wiley Series in Probability and Statistics. New York: John Wiley & Sons, 1968

    work page 1968

  11. [11]

    J. R. Norris,Markov Chains. Cambridge, UK: Cambridge University Press, 1997

    work page 1997

  12. [12]

    S. M. Ross,Introduction to Probability Models, 12th ed. London, UK: Academic Press (Elsevier), 2020, includes new coverage on coupling, renewal theory, queueing theory, and Poisson processes; winner of a 2020 Textbook Excellence Award (Texty). 9

    work page 2020

  13. [13]

    Frequency noise characterization of diode lasers for vapor- cell clock applications,

    G. Antona, M. Gozzelino, S. Micalizio, C. E. Calosso, G. A. Costanzo, and F. Levi, “Frequency noise characterization of diode lasers for vapor- cell clock applications,”IEEE Transactions on Instrumentation and Measurement, vol. 71, pp. 1–8, 2022

    work page 2022

  14. [14]

    Chip-based laser with 1-hertz integrated linewidth,

    J. Guo, C. A. McLemore, C. Xiang, D. Lee, L. Wu, W. Jin, M. Kelleher, N. Jin, D. Mason, L. Chang, A. Feshali, M. Paniccia, P. T. Rakich, K. J. Vahala, S. A. Diddams, F. Quinlan, and J. E. Bowers, “Chip-based laser with 1-hertz integrated linewidth,”Science Advances, vol. 8, no. 43, p. eabp9006, 2022. [Online]. Available: https://www.science.org/doi/abs/10...

    work page doi:10.1126/sciadv.abp9006 2022

  15. [15]

    Ultra-precise optical-frequency stabilization with heterogeneous iii–v/si lasers,

    L. Stern, W. Zhang, L. Chang, J. Guo, C. Xiang, M. A. Tran, D. Huang, J. D. Peters, D. Kinghorn, J. E. Bowers, and S. B. Papp, “Ultra-precise optical-frequency stabilization with heterogeneous iii–v/si lasers,”Opt. Lett., vol. 45, no. 18, pp. 5275–5278, Sep 2020. [Online]. Available: https://opg.optica.org/ol/abstract.cfm?URI=ol-45-18-5275

    work page 2020

  16. [16]

    Frequency and intensity noise of single frequency fiber bragg grating lasers,

    E. Rønnekleiv, “Frequency and intensity noise of single frequency fiber bragg grating lasers,”Optical Fiber Technology, vol. 7, no. 3, pp. 206–235, 2001. [Online]. Available: https://www.sciencedirect.com/ science/article/pii/S1068520001903578

    work page 2001

  17. [17]

    Laser frequency noise characterization using high-finesse plano–concave optical microresonators,

    D. Martin-Sanchez, E. Z. Zhang, J. Paterson, J. A. Guggenheim, Z. Liu, and P. C. Beard, “Laser frequency noise characterization using high-finesse plano–concave optical microresonators,” Opt. Lett., vol. 49, no. 3, pp. 678–681, Feb 2024. [Online]. Available: https://opg.optica.org/ol/abstract.cfm?URI=ol-49-3-678

    work page 2024

  18. [18]

    Hertz-linewidth semiconductor lasers using cmos-ready ultra-high-q microresonators,

    W. Jin, Q.-F. Yang, L. Chang, B. Shen, H. Wang, M. A. Leal, L. Wu, M. Gao, A. Feshali, M. Paniccia, K. J. Vahala, and J. E. Bowers, “Hertz-linewidth semiconductor lasers using cmos-ready ultra-high-q microresonators,”Nature Photonics, vol. 15, no. 5, pp. 346–353, May

    work page

  19. [19]

    Available: https://doi.org/10.1038/s41566-021-00761-7

    [Online]. Available: https://doi.org/10.1038/s41566-021-00761-7

    work page doi:10.1038/s41566-021-00761-7

  20. [20]

    Yariv and P

    A. Yariv and P. Yeh,Photonics: Optical Electronics in Modern Com- munications, 6th ed. Oxford University Press, 2007

    work page 2007

  21. [21]

    Microring resonator channel dropping filters,

    B. Little, S. Chu, H. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,”Journal of Lightwave Technology, vol. 15, no. 6, pp. 998–1005, 1997

    work page 1997

  22. [22]

    Okamoto,Fundamentals of optical waveguides, 2nd ed

    K. Okamoto,Fundamentals of optical waveguides, 2nd ed. Academic Press, May 2014

    work page 2014

  23. [23]

    Rubiola,Phase Noise and Frequency Stability in Oscillators, ser

    E. Rubiola,Phase Noise and Frequency Stability in Oscillators, ser. The Cambridge RF and Microwave Engineering Series. Cambridge University Press, 2008

    work page 2008

  24. [24]

    D. A. Levin, Y . Peres, and E. L. Wilmer,Markov chains and mixing times. American Mathematical Society, 2006. [Online]. Available: http://scholar.google.com/scholar.bib?q=info:3wf9IU94tyMJ:scholar. google.com/&output=citation&hl=en&as sdt=2000&ct=citation&cd=0

    work page 2006

  25. [25]

    Delta functions,

    H.-H. Kuo, “Delta functions,” inWhite Noise Distribution Theory. CRC Press, May 2018, pp. 61–76

    work page 2018

  26. [26]

    J. W. Goodman,Speckle phenomena in optics, 2nd ed., ser. Press Monographs. Bellingham, W A: SPIE Press, May 2025

    work page 2025