Why brighter light also shakes the mirror harder

Set displacement and time scales

Use $q=x/(\sqrt2x_{\rm zpf})$ with conjugate $p$ satisfying $[q,p]=i$ . Both ground-state variances are $1/2$ . Set $\tau=\Omega_mt$ and $\nu=\omega/\Omega_m$ . Measurement strength $k$ and damping $\gamma$ are normalized by mechanical frequency.

The continuous-measurement model uses a weak amplitude-damping bath whose memory is short on the observed time scale, the Markov approximation. Both quadratures decay slowly and receive equal thermal diffusion. It joins the viscous model of Chapter 2 near resonance at high Q, but their exact far-off-resonant spectra differ.

A=\begin{pmatrix}-\gamma/2&1\\-1&-\gamma/2\end{pmatrix},\qquad D_{\rm th}=\gamma(n_{\rm th}+1/2)I,\qquad D_{\rm ba}=\begin{pmatrix}0&0\\0&k\end{pmatrix}

The drift matrix evolves the phase-space centre. Diffusion adds covariance per unit normalized time. Position-measurement backaction diffuses momentum.

One strength sets both noises

Let $\eta$ be total detection efficiency from interaction to saved record, between zero and one. The record increment $dZ$ contains accumulated position signal and a random increment $dW$ of zero mean and variance $d\tau$ . This is the limit of many independent short-time fluctuations.

Define $R=1/(4\eta k)$ . Position-equivalent readout noise has two-sided spectrum $R$ ; white force noise in the momentum equation has spectrum $k$ . Losing information downstream does not undo the photons’ mechanical kicks.

dZ=\langle q\rangle_c\,d\tau+\sqrt R\,dW,\qquad S_{qq}^{\rm imp}=\frac1{4\eta k},\qquad S_{FF}^{\rm ba}=k

These are two-sided symmetrized spectra per normalized angular frequency, with the same two-pi integration measure as before. The master-equation coefficient multiplies the position-measurement dissipator.

Source locations: Jacobs & Steck

Pass force noise through the mechanical response

Set $a=\gamma/2$ . Force in the normalized momentum equation reaches position through $\chi_{qF}$ . With no feedback, resonant phase readout and no extra noise correlations, the measured spectrum adds thermal motion, backaction motion and imprecision.

Detuned readout, squeezed input or another detection quadrature can correlate imprecision with backaction. Then include a cross spectrum. The standard quantum limit describes a specified measurement arrangement.

\chi_{qF}(\nu)=\frac1{(a-i\nu)^2+1},\qquad S_{ZZ}=S_{qq}^{\rm th}+k|\chi_{qF}|^2+\frac1{4\eta k}

The thermal term follows the same drift and diffusion model. A white readout background has divergent infinite-bandwidth variance; record variance requires a stated finite bandwidth.

S_{\rm imp}=1/(4k)

S_{\rm ba}=k

S_{\rm add}

Calculated added noise at unit susceptibility and efficiency. Imprecision falls inversely with strength; backaction rises linearly. Their sum has a minimum. Both axes use logarithmic scaling.

Source locations: Aspelmeyer

Find what the optimum optimizes

At fixed frequency, differentiate the added noise with respect to $k$ . Backaction grows linearly and imprecision falls inversely. At the stationary point the two contributions are equal.

Worked example: at a frequency where $|\chi_{qF}|=2$ and efficiency is one, choose $k=0.25$ . Each contribution is one, so added noise is two. At efficiency $1/4$ , optimal strength becomes $0.5$ . Each contribution is then two, giving four in total.

This optimizes displacement-equivalent noise at one frequency. Estimating an entire waveform, a force pulse or a mechanical state uses information across the record. That is the next chapter’s task.

k_{\rm best}=\frac1{2\sqrt\eta|\chi_{qF}|},\qquad S_{\rm add,min}=\frac{|\chi_{qF}|}{\sqrt\eta}

The two added noises are assumed uncorrelated. Cooling transition rates require the positive–negative frequency imbalance of an unsymmetrized quantum force spectrum.

Predict, change, compare, explain

Sweep from too little light to too much

Predict which noise dominates each extreme, then sweep strength continuously.
Reduce efficiency and compare imprecision with backaction. Explain their different responses.
Save the record and its sample rate. Compare the spectral optimum with how clearly motion can be resolved over time.

Exercises with explanations

Efficiency falls fourfold at fixed optical strength. Does backaction decrease?

Read the explanation

No in this model. Interaction with the oscillator sets backaction; detection loss removes information afterward. Imprecision grows fourfold and backaction force noise stays fixed.

Does a spectrum below a former background establish that every quantum limit has been surpassed?

Read the explanation

First specify the observable, spectral convention, correlations, efficiency and calibration. A specialized readout can surpass the limit of uncorrelated position measurement and still satisfy the full quantum noise relation.

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.

Aspelmeyer, Kippenberg & Marquardt, Cavity Optomechanics, arXiv:1303.0733v1 (2013); RMP 86, 1391 (2014)

Pinned arXiv v1: II.1–II.2 cavity linewidth and mechanical spectra; III.1–III.3 radiation pressure and linearization; V.2 and VII.1 dynamical backaction and cooling; VI.1–VI.2 measurement and single-quadrature readout; X.6 nonlinear quantum optomechanics. Equations and examples here are derived in the course convention. This is a review, not a new experimental dataset.

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.