Turn an observed motion into a cooling force

Use only the record that has arrived

An experimental controller can access present and past photocurrent. A simulator also stores phase-space samples. Feeding their hidden momentum directly into control would grant information that the experiment lacks. The filter must estimate motion from the saved observation before the controller uses that estimate.

Keep the normalized units. Force $u$ enters the momentum equation and nonnegative gain $g_{\rm fb}$ sets $u=-g_{\rm fb}\mu_p$ . A discrete implementation selects force from the estimate already available, holds it for one interval, then processes the next observation.

u_j=-g_{\rm fb}\mu_p(\tau_j^-),\qquad \dot{\boldsymbol\mu}_c=A\boldsymbol\mu_c+e_pu+\text{measurement update},\quad e_p=\begin{pmatrix}0\\1\end{pmatrix}

The minus superscript denotes information available before the force is applied. New observations can affect subsequent force intervals.

Source locations: Rossi et al.

Check the work done by the force

Dimensionless energy is $E=(q^2+p^2)/2$ ; multiply by $\hbar\Omega_m$ for physical energy. Force supplies power $up$ . Conditional expectation of momentum is $\mu_p$ , so ideal instantaneous damping removes mean power $g_{\rm fb}\mu_p^2$ .

Worked example: the filter reports $\mu_p=2$ and gain is $0.5$ . Force is $u=-1$ and expected feedback power is $-2$ in normalized units. If conditional momentum variance is $0.7$ , this known force translates the state; it does not eliminate that variance.

\mathbb E[up\mid Z_{\leq\tau}]=-g_{\rm fb}\mu_p^2,\qquad \bar n=\frac{\operatorname{tr}V+|\boldsymbol\mu_c|^2-1}{2}

Occupation includes conditional width and energy of the centre. The conversion requires physical covariance and the stated quadrature normalization.

Bring the centres together

Given a record, a known linear force shifts the centre without changing the conditional covariance equation. Across records, feedback can reduce centre variation and therefore unconditional energy. Total covariance explains how the two statements fit together.

With poor efficiency the conditioned state remains broad. Even perfect removal of its centre would leave this unknown motion. Gain cannot replace detection efficiency; final energy depends on both estimation and the thermal bath.

In the ideal continuous linear Gaussian model, known-force state estimation and linear control can be designed separately. This separation principle relies on a correct model and causal observations. Saturation, delay and actuator noise change the result.

Source locations: Rossi et al.

Delay can turn damping into driving

After half a mechanical period the motion has reversed. A force based on an old velocity can then push with the motion. Larger gain increases the energy supplied at the wrong phase, so experimental controllers require calibrated delay, bandwidth and actuator response.

When comparing feedback settings, keep measurement strength, efficiency, seed and initial state fixed. Recompute stored forces from earlier observations. Feedback correlations can also suppress the in-loop photocurrent spectrum, called noise squashing. An independent temperature estimate needs another probe or a full treatment of these correlations.

Predict, change, compare, explain

Compare feedback with the same readout chain

Hold measurement strength and efficiency fixed. Predict separately the effects on conditional width and centre motion, then switch feedback.
Save observations and applied force. Replay the filter and verify the order of estimation and each held-force interval.
Repeat at lower efficiency. Explain residual energy as conditional width plus centre variation.

Exercises with explanations

A simulation uses sample momentum directly and cools more strongly than the record-driven filter. Is that an experimental prediction?

Read the explanation

It is an extra-information diagnostic. A realizable prediction must use the noisy observations, known parameters and past controls available to a real controller.

Can unconditional energy decrease with unchanged conditional covariance?

Read the explanation

Yes. Total energy also contains variation of conditional centres between records. Feedback can reduce that variation without narrowing the state conditioned on one record.

Read the sources

Derivations, worked examples and explanatory figures are authored for the stated models. Synthetic records and model references are labelled separately from experimental results. The source locations below identify the reviewed theory and the limited numerical comparison.

Rossi et al., Observing and Verifying the Quantum Trajectory of a Mechanical Resonator, arXiv:1812.00928v1 (2018)

Eqs. 1–5 and Figs. 1–3: rotating-frame Gaussian conditioning and verification using a backward estimate. The course derives a laboratory-frame two-quadrature example. Its synthetic records are separate from the experiment and do not reproduce its device parameters.

Rossi et al., Measurement-based quantum control of mechanical motion, arXiv:1805.05087v2 (2018)

Published in Nature 563, 53–58 (2018). Experimental feedback cooling from an optical position record. Our causal held-force controller is an instructional Gaussian model, distinct from the experimental filter and electronics.