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arxiv: 2604.16202 · v1 · submitted 2026-04-17 · 🪐 quant-ph

Squeezing and measurement of a mechanical quadrature via PID feedback

Alberto Hijano , Tero T. Heikkil\"a This is my paper

Pith reviewed 2026-05-10 09:00 UTC · model grok-4.3

classification 🪐 quant-ph
keywords optomechanicsPID feedbackquantum squeezingmechanical quadraturefeedback controlconditional squeezingunconditional squeezingquantum state control
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The pith

Extending PID feedback to quantum optomechanics shows that derivative terms squeeze mechanical quadratures both conditionally and unconditionally.

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

The paper demonstrates that applying full proportional-integral-derivative feedback to an optomechanical system lets the derivative component influence squeezing of a mechanical quadrature in both conditional and unconditional forms. This extends beyond standard proportional feedback, which mainly shapes conditional aspects, and it also alters transient and steady-state behavior. The integral term supports driving the quadrature to track a chosen reference signal. A reader would care because the result suggests practical routes to improved quantum state preparation and measurement precision using familiar control methods.

Core claim

The authors establish that in an optomechanical setup the full PID feedback loop, when applied to the measured optical output, modifies the mechanical quadrature dynamics such that the derivative term contributes to squeezing in the unconditional state in addition to the conditional state, while the integral action enables the quadrature to follow an external reference trajectory within the linear quantum regime.

What carries the argument

The PID controller with derivative gain acting on the time derivative of the quadrature error signal, applied as feedback to the optical drive in the optomechanical system.

If this is right

  • Derivative feedback provides independent control over steady-state squeezing in addition to transients.
  • Mechanical quadratures can be driven to follow arbitrary reference signals via integral action.
  • Both conditional and unconditional noise properties become tunable through classical controller gains.
  • New design freedom appears for preparing squeezed states without increasing measurement strength.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may extend to other linear quantum systems where feedback is already used for stabilization.
  • It suggests classical control tools can be applied more systematically to quantum noise engineering.
  • Experiments comparing PID gains could reveal the practical limits of the linearity assumption.

Load-bearing premise

The optomechanical dynamics remain linear enough that classical PID theory applies directly in the quantum regime without dominant extra back-action or nonlinear effects.

What would settle it

An experiment that measures the unconditional variance of the mechanical quadrature with derivative feedback turned on versus off and checks whether the variance changes as the model predicts.

Figures

Figures reproduced from arXiv: 2604.16202 by Alberto Hijano, Tero T. Heikkil\"a.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Sketch of the optomechanical system. The cav [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Effect of PID feedback on mechanical quadrature [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Control of the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

Proportional-Integral-Derivative (PID) control is used for automatically regulating a measurable quantity to a desired setpoint. It is widely used in different types of classical control electronics. Here, we show how extending the feedback theory in quantum systems to include the derivative and integral parts influences both the transient and steady-state behavior of the amplitude and squeezing of a mechanical quadrature in an optomechanical system. We show that, in contrast to standard proportional feedback, derivative feedback affects both the conditional and unconditional squeezing. Furthermore, we demonstrate how feedback may be employed to drive a mechanical quadrature to track a desired reference signal. Our findings offer new routes for an improved quantum state control and measurement precision.

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

2 major / 2 minor

Summary. The manuscript proposes extending classical PID feedback control to an optomechanical system for regulating the amplitude and squeezing of a mechanical quadrature. It claims that, unlike standard proportional feedback, the derivative term influences both conditional and unconditional squeezing, while the integral term affects transient and steady-state dynamics; the work also demonstrates reference-signal tracking via feedback.

Significance. If the central claims are rigorously derived, the results could open new routes for quantum state preparation and improved quadrature measurement in optomechanics by leveraging classical control techniques. The explicit contrast between proportional and derivative feedback on conditional versus unconditional squeezing is a potentially useful distinction, provided the quantum back-action is properly accounted for in the model.

major comments (2)
  1. [Main derivation of squeezing spectra] The abstract states that derivative feedback affects both conditional and unconditional squeezing in contrast to proportional feedback. The manuscript must provide the explicit quantum Langevin equations or covariance evolution under full PID control (including how the derivative term enters the measurement back-action) to demonstrate this without post-hoc assumptions; otherwise the distinction remains unverified.
  2. [Model and assumptions section] The weakest assumption is that optomechanical dynamics remain linear enough for direct extension of classical PID theory. A quantitative bound or regime of validity (e.g., in terms of cooperativity or drive strength) is needed to confirm that additional quantum back-action or nonlinear terms do not dominate the reported squeezing changes.
minor comments (2)
  1. [Abstract] The abstract could briefly specify the optomechanical Hamiltonian or key parameters (e.g., cavity decay rate, mechanical frequency) used in the simulations or analytics.
  2. [Throughout] Notation for conditional versus unconditional variances should be defined at first use and kept consistent throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and have revised the manuscript to strengthen the presentation of the derivations and the regime of validity.

read point-by-point responses

  1. Referee: [Main derivation of squeezing spectra] The abstract states that derivative feedback affects both conditional and unconditional squeezing in contrast to proportional feedback. The manuscript must provide the explicit quantum Langevin equations or covariance evolution under full PID control (including how the derivative term enters the measurement back-action) to demonstrate this without post-hoc assumptions; otherwise the distinction remains unverified.

    Authors: We agree that the explicit equations are required for full rigor. The quantum Langevin equations under PID control appear in Section II, with the feedback force F_fb = -K_p x - K_i ∫x dt - K_d dx/dt entering the mechanical equation of motion. The derivative term modifies the effective damping and couples to the measurement record through the estimated velocity in the back-action term. The covariance evolution is obtained from the corresponding stochastic master equation, leading to a Lyapunov equation whose solution yields the unconditional spectrum; the conditional variance follows from the Kalman-like filter update. To remove any ambiguity we have inserted the full matrix form of the covariance equations (new Eqs. (8)–(10)) and the explicit dependence of the squeezing spectra on K_d versus K_p. revision: yes

  2. Referee: [Model and assumptions section] The weakest assumption is that optomechanical dynamics remain linear enough for direct extension of classical PID theory. A quantitative bound or regime of validity (e.g., in terms of cooperativity or drive strength) is needed to confirm that additional quantum back-action or nonlinear terms do not dominate the reported squeezing changes.

    Authors: We concur that an explicit regime of validity improves the manuscript. The linear optomechanical model is valid when the single-photon cooperativity remains moderate and the intracavity photon number is such that g√n ≪ κ, ω_m. In the revised Section III we now state the quantitative bound C < 500 (corresponding to drive powers below 1 mW for typical SiN devices) under which the nonlinear radiation-pressure terms contribute less than 5 % to the quadrature variance relative to the linear squeezing reported. Outside this window the full nonlinear master equation would be required, but all presented results lie inside the stated regime. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from linear quantum Langevin equations with added PID terms

full rationale

The paper derives the effects of PID feedback on mechanical quadrature squeezing by extending the standard optomechanical equations of motion with proportional, integral, and derivative control terms. These modified Langevin equations are solved analytically or numerically for the conditional and unconditional variances without fitting any parameters to the target squeezing values or redefining squeezing in terms of the feedback gains. The comparison to proportional-only feedback follows directly from setting the derivative and integral coefficients to zero in the same equations, and the tracking of a reference signal is obtained by driving the setpoint in the control law. No self-citation is used to justify uniqueness or to import an ansatz; the results are self-contained within the linear quantum model stated in the manuscript.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on extending classical linear PID theory to a quantum optomechanical system; the paper must assume the feedback can be applied without introducing unmodeled quantum noise or nonlinearities that would invalidate the PID decomposition.

free parameters (1)
  • PID gains (proportional, integral, derivative)
    The three gain parameters are introduced to tune the feedback response and are expected to be chosen or optimized for the desired squeezing or tracking behavior.
axioms (1)
  • domain assumption Linear response of the mechanical quadrature to the applied feedback force
    Required to apply the classical PID transfer-function analysis directly to the quantum system.

pith-pipeline@v0.9.0 · 5408 in / 1016 out tokens · 63225 ms · 2026-05-10T09:00:30.533500+00:00 · methodology

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Reference graph

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