Why the same mirror has two different widths

Begin with one finite-precision observation

Suppose position has a Gaussian prior with mean $\mu$ and variance $V$ . The detector returns $y=q+\epsilon$ , with independent error variance $N$ . Bayes’ rule multiplies the prior by the likelihood: the chance of obtaining that readout at each possible position.

Add the two Gaussian exponents and complete the square. The posterior weights a sharp observation more strongly and retains more of a sharp prior when the detector is noisy. This calculation uses the readout and its error model.

V_{\rm new}=\left(V^{-1}+N^{-1}\right)^{-1},\qquad \mu_{\rm new}=\mu+\frac V{V+N}(y-\mu)

This scalar update illustrates information gain. Quantum position measurement must also add the corresponding momentum backaction.

Take many small updates

Write $\boldsymbol r=(q,p)^T$ , with conditional mean $\boldsymbol\mu_c$ and symmetrized covariance $V$ . Vector $e=(1,0)^T$ selects position. Keep $A$ , $D=D_{\rm th}+D_{\rm ba}$ and $R$ from Chapter 5.

A short-time average has detector variance $R/d\tau$ . Expand the Gaussian update to first order in time and add free evolution and diffusion. The observation minus its prediction is the innovation. Only this newly acquired information shifts the estimate.

d\boldsymbol\mu_c=A\boldsymbol\mu_c\,d\tau+\frac{Ve}{R}\left(dZ-e^T\boldsymbol\mu_c\,d\tau\right)

Compute the innovation from the saved record. The filter has no input for the simulator’s hidden position or momentum.

\dot V=AV+VA^T+D-\frac{Vee^TV}{R}

Evolution and diffusion supply the first terms; information removes uncertainty through the final term. Such a quadratic covariance equation is called a Riccati equation.

Source locations: Jacobs & Steck · Rossi et al.

Narrow states can have widely scattered centres

Mix the conditional states from many independent records to recover the state obtained when the record is ignored. Its width combines uncertainty inside each conditional state and variation between their centres. Plotting only the first can make localization look like cooling of the entire ensemble.

Worked example without dynamics: take prior mean zero, variance $V=2$ and detector variance $N=1$ . Reading $y=1$ gives mean $2/3$ and variance $2/3$ . Across repetitions the readout variance is three, so posterior means have variance $(2/3)^2\times3=4/3$ . The sum restores two.

V_{\rm uncond}=\mathbb E[V_c]+\operatorname{Cov}(\boldsymbol\mu_c)

This is total covariance. With feedback absent, averaged conditional moments must recover the same unconditional master equation.

q

p

Illustration of total covariance. Each small ellipse is one conditioned state. Their centres vary between records. The broad outline represents their mixture; these are explanatory Gaussian distributions, not observed samples.

Interpret the trajectory

A conditional quantum trajectory is the evolution of $\rho_c$ given the observation history. It does not assign the mirror a hidden path with exact simultaneous position and momentum. A Gaussian state has a positive Wigner function; phase-space samples can generate the same observation statistics and serve as a numerical representation.

A physical covariance must also satisfy $\det V\geq1/4$ . Narrowing position requires accounting for the conjugate component, correlations and lost information. A finite-step integrator should preserve this condition and converge as its step decreases.

Rossi and colleagues checked their forward estimates against estimates inferred from later observations. Validating the state represented by a filter requires this additional statistical comparison.

Source locations: Rossi et al.

Predict, change, compare, explain

Track a state using the record alone

Predict how efficiency changes the conditional ellipse. Compare efficiencies with feedback off.
Save and replay a record. Check that the filter reads the observation column and keeps simulator phase-space samples separate.
Compare conditional width, centre variation and unconditional width using total covariance.

Exercises with explanations

In the example, what changes if the readout is negative one?

Read the explanation

Posterior variance stays two thirds because the information in a linear Gaussian observation depends on its precision. The posterior mean becomes negative two thirds.

Stop recording while leaving the same light on. What happens to localization and backaction?

Read the explanation

Available information vanishes, so the conditioned estimate spreads toward unconditional uncertainty. Light still interacts with the oscillator and backaction diffusion continues.

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.

Jacobs & Steck, A straightforward introduction to continuous quantum measurement, arXiv:quant-ph/0611067v1 (2006)

Continuous position measurement, stochastic master equations and Gaussian moment equations. Our strength multiplies the position-measurement dissipator; a double-commutator convention can differ by a factor of two. The course defines its record and diffusion explicitly.

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.