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.

How closely can light watch?

Drag the end mirror to displace it. Turn up the light, then let the detector record drive the feedback.

Measurement strength

0.12

k/\Omega_{\mathrm m}

Detection efficiency

0.8

\eta

Bath occupation

1

n_{\mathrm{th}}

Record-driven damping

0

g_{\mathrm{fb}}/\Omega_{\mathrm m}

Preparation and model

Initial displacement

0

\langle q(0)\rangle

Random seed

$[q,p]=i,\quad V_{\mathrm{vac}}=\tfrac12 I,\quad \gamma/\Omega_{\mathrm m}=0.16$

A resonant broadband cavity is eliminated. Feedback acts on the next sample from the estimated momentum. Mirror motion is a simulated Wigner sample.

Jacobs and Steck · continuous measurement

Schematic geometry · motion magnified and amplitude compressed to stay in frame · synthetic record replay

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 recordConditional meanWigner 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 + vacuumBackactionImprecision

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

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 stateWhole 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

Study quantum optomechanics from moving mirrors to single photons

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 misconceptionThe inferred ellipse shrank, so the mirror must have lost energy.

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

Run the experiment

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.
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.
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.

Continuous Optomechanical Measurement

The noisy light and the inferred motion

Where the measurement noise comes from

Knowing more and moving less

Follow the mirror through a noisy record

Core mathematical model

A calibrated detector record

The two added noises

Information and feedback play different roles

Common difficulties

A small conditional state need not mean a cold ensemble

Run the experiment

Turn up the light

Cool using the same samples

Lose some of the photons

Related experiments and articles

From a moving mirror to quantum measurement