---
title: "Continuous Optomechanical Measurement Interactive Lab | Sandbox Physics"
description: "Why does brighter light reveal a mirror’s motion and shake it more?"
canonical_url: "https://sandboxphysics.com/labs/optomechanical-continuous-measurement/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/optomechanical-continuous-measurement/"
---

FR12 · A mirror watched by light

# Continuous Optomechanical Measurement

**Why does brighter light reveal a mirror’s motion and shake it more?** More probe photons reduce phase shot noise, but their fluctuating pressure adds momentum noise. A filter learns from the detected light; feedback uses that estimate to damp the mirror. Lost photons still disturb it.

Turn up the probe and compare the detector record with the inferred motion. Use that same record to cool a vibrating mirror, then compare one conditional state with an ensemble of experiments.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

A resonant, broadband cavity is eliminated to obtain one Markovian Gaussian mechanical mode. Position and momentum have vacuum variance one half; time is scaled by the mechanical angular frequency. The thermal bath uses weak-damping rotating-wave Lindblad operators. The balanced homodyne detector reads the phase quadrature; detection loss reduces information without reducing radiation-pressure disturbance. Feedback is a causal force from the record-based momentum estimate. Plotted spectra are double-sided, symmetrized, open-loop model references. Detector samples are synthetic. A moving mirror replays a positive Wigner phase-space sample, not a measured quantum trajectory. Geometry is schematic; motion is magnified and its amplitude compressed to stay in frame. Quantitative curves and readouts use the uncompressed model coordinates. Optical delay, finite cavity bandwidth, technical laser noise, actuator saturation and non-Gaussian states are omitted.

State widths below refer to the end of the stored record.

Within one conditional state

$n_{\mathrm c}=\tfrac12(\operatorname{tr}V_{\mathrm c}-1)$

$0$

Across repeated experiments

$n_{\mathrm{ens}}=\tfrac12(\operatorname{tr}V_{\mathrm{ens}}-1)$

$0$

Conditional position uncertainty

$\sqrt{V_{qq,\mathrm c}}$

$0$

Open-loop resonance optimum

$k_{\mathrm{opt}}/\Omega_{\mathrm m}$

$0$

## The noisy light and the inferred motion

Detector record Conditional mean Wigner simulator sample

$\tau=\Omega_{\mathrm m}t,\qquad q=x/(\sqrt2 x_{\mathrm{zpf}})$

The amber curve averages ten stored detector samples per displayed point. The filter uses every original sample. Gray is the simulated phase-space sample, not measured motion.

## Where the measurement noise comes from

Thermal + vacuum Backaction Imprecision

$\omega/\Omega_{\mathrm m},\qquad S_{qq}^{\mathrm{sym}}(\omega)$

Open-loop theoretical references. Vertical axis is logarithmic. The integral over both frequency signs, divided by two pi, gives variance; imprecision is the white detector floor.

## Knowing more and moving less

One conditional state Whole ensemble

$q,\quad p;\qquad V_{\mathrm{ens}}=V_{\mathrm c}+\operatorname{Cov}(\boldsymbol\mu_{\mathrm c})$

Ellipses show one-standard-deviation covariance contours, both recentered to compare their widths. Dots sample the distribution of conditional means from the same Gaussian model. Feedback narrows that distribution.

What is stored and what is a reference

CSV keeps all detector samples, causal commands, conditional means and covariances, the seed, time step, parameters and version. Wigner simulator coordinates have separate columns. Replaying the detector samples alone reproduces the filter and commands. The spectrum is an analytic open-loop reference; the ensemble covariance is propagated for the same finite-time measurement channel and feedback law.

[Aspelmeyer, Kippenberg and Marquardt · measurement and feedback](https://arxiv.org/html/1303.0733#S6)

[Study quantum optomechanics from moving mirrors to single photons](https://sandboxphysics.com/stories/quantum-optomechanics/)

Physics tutorial

## Follow the mirror through a noisy record

**Background** Turn up the probe and compare the detector record with the inferred motion. Use that same record to cool a vibrating mirror, then compare one conditional state with an ensemble of experiments.

**Why it matters** More light supplies more information and more random force. Their competition matters before a feedback controller can help.

Start with the essentials

- **Focus question**: Can observing a mirror more precisely cool its actual motion?
- **One-sentence intuition**: Conditioning narrows what one record leaves uncertain. Feedback changes the distribution across repeated experiments.

### Core mathematical model

#### A calibrated detector record

$$
\mathrm d y=\langle q\rangle_{\mathrm c}\,\mathrm d\tau+\frac{\mathrm dW}{2\sqrt{\eta k}}
$$

Dimensionless position and momentum have vacuum variance one half. Normalized time is mechanical angular frequency times time. The Wiener increment has variance equal to the time increment.

#### The two added noises

$$
S_{qq}^{\mathrm{imp}}=\frac{1}{4\eta k},\qquad S_{FF}^{\mathrm{BA}}=k,\qquad S_{qq}^{\mathrm{imp}}S_{FF}^{\mathrm{BA}}=\frac{1}{4\eta}
$$

These are symmetrized double-sided spectra per normalized angular frequency. Force noise must pass through the mechanical susceptibility before it becomes displacement noise. The reference spectrum always assumes feedback off.

#### Information and feedback play different roles

$$
V_{\mathrm{ens}}=V_{\mathrm c}+\operatorname{Cov}(\boldsymbol\mu_{\mathrm c}),\qquad u_j=-g_{\mathrm{fb}}\widehat p_{j-1}
$$

The ensemble covariance includes both the uncertainty within each conditional state and the spread of its centers. The next force uses the momentum estimated from earlier detector samples.

### Common difficulties

#### A small conditional state need not mean a cold ensemble

**Typical misconception** The inferred ellipse shrank, so the mirror must have lost energy.

**Better mental model** Different records place narrow states at different centers. Feedback damps those estimated centers; conditioning alone does not remove the ensemble spread.

### Run the experiment

1. 01

   #### Turn up the light

   Compare Light barely watches, Resolve the motion and Light shakes the mirror.

   **What to observe:** The white imprecision floor falls. The backaction peak grows, and the mirror’s simulated motion broadens.

2. 02

   #### Cool using the same samples

   Choose Cool from the record, save the run, and set feedback to zero. Restore the saved run and export its samples.

   **What to observe:** Feedback reduces the amber ensemble ellipse. The cyan conditional ellipse describes the remaining uncertainty for one record. Exported forces can be rebuilt from exported detector samples.

3. 03

   #### Lose some of the photons

   Lower Detection efficiency at fixed Measurement strength and compare the two state widths.

   **What to observe:** The detector learns less, but the mirror still receives the same backaction. Feedback then has less reliable information to act on.

Physical conclusion

A continuous measurement supplies a noisy record and a physical disturbance. A causal controller can use the first to reduce motion caused by the second and by the thermal bath.

## Related experiments and articles

Quantum optomechanics · Reading path

### From a moving mirror to quantum measurement

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Read a mechanical temperature from a noisy trace**](https://sandboxphysics.com/stories/quantum-optomechanics/spectra-and-calibration/)
2. [Article **Why brighter light also shakes the mirror harder**](https://sandboxphysics.com/stories/quantum-optomechanics/imprecision-and-backaction/)
3. [Article **Why the same mirror has two different widths**](https://sandboxphysics.com/stories/quantum-optomechanics/conditional-state/)
4. [Article **Turn an observed motion into a cooling force**](https://sandboxphysics.com/stories/quantum-optomechanics/feedback-from-records/)
5. [Article **Where backaction goes when you watch one quadrature**](https://sandboxphysics.com/stories/quantum-optomechanics/backaction-evasion-and-mechanical-states/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)