Read a mechanical temperature from a noisy trace

Ask how the same oscillator responds at each frequency

For mass $M$ and frequency $\Omega_m$ , let $\Gamma_m$ denote energy decay rate. A viscous force opposes velocity. At each drive frequency, multiply force by the mechanical susceptibility $\chi_m$ to obtain displacement. Its units are metres per newton.

Near resonance, repeated pushes accumulate motion. Away from resonance their timing slips relative to the oscillator. Peak position largely reports inertia and elasticity; width reports energy loss.

M(\ddot x+\Gamma_m\dot x+\Omega_m^2x)=F,\qquad \chi_m(\omega)=\frac{1}{M(\Omega_m^2-\omega^2-i\Gamma_m\omega)}

Fourier components have negative time exponents. In weak damping, the displacement envelope decays at half the energy decay rate.

Source locations: Aspelmeyer

Spectral area gives variance

Subtract the mean displacement. The two-sided spectrum $S_{xx}^{(2)}(\omega)$ includes positive and negative angular frequencies. Its correlation function asks how similar the motion remains after a time delay. Fourier transformation distributes that correlation over frequency. For quantum displacement, use the symmetrized correlation, averaging the two operator orders.

Many instruments show a one-sided spectrum per hertz: retain positive frequencies and add the equal contribution from negative frequencies. The equations specify our normalization. Integrating over a finite band converts square metres per hertz to square metres.

\begin{aligned}S_{xx}^{(2)}(\omega)&=\int_{-\infty}^{\infty}\!C_{xx}(t)e^{i\omega t}\,dt,\\ \langle x^2\rangle&=\int_{-\infty}^{\infty}\!\frac{d\omega}{2\pi}S_{xx}^{(2)}(\omega)=\int_0^\infty\!df\,S_{xx}^{(1)}(f),\\ S_{xx}^{(1)}(f)&=2S_{xx}^{(2)}(2\pi f),\qquad f>0.\end{aligned}

The angular-frequency integration measure contains two pi. Some sources absorb that factor into the spectrum instead; convert definitions before comparing numbers.

Use equilibrium to set the area

In the high-temperature white-noise limit, a viscous bath both removes energy and kicks the oscillator. At equilibrium the two effects balance. Multiply the force spectrum by susceptibility magnitude squared, then integrate. It must recover the equipartition variance, providing an independent area check.

Worked example: choose $M=10^{-12}\,\mathrm{kg}$ , $\Omega_m/(2\pi)=1\,\mathrm{MHz}$ and $T=300\,\mathrm K$ . Equipartition gives a displacement standard deviation of $10.2\,\mathrm{pm}$ . Suppose the converted mechanical peak has area $2.60\times10^{-23}\,\mathrm{m^2}$ . If mass and temperature are reliable, the displacement-amplitude calibration is too small by two: its variance is too small by four.

S_{FF}^{(2)}=2M\Gamma_mk_BT,\quad S_{xx}^{(2)}=|\chi_m|^2S_{FF}^{(2)},\quad \langle x^2\rangle=\frac{k_BT}{M\Omega_m^2}

This is a classical bath. At low temperature retain zero-point fluctuations; a high-Q quantum mode has variance equal to zero-point variance times twice the thermal occupation plus one.

Source locations: Aspelmeyer · Callen & Welton

The detector adds its own fluctuations

The instrument sees the whole readout chain. A flat optical background, a mechanical peak and electronic noise may all contribute. Calibrate photocurrent response to displacement independently. Any subtraction of background under the peak needs a stated model. Counting that background as motion overestimates temperature.

A finite observation also broadens spectral lines. Frequencies above half the sample rate can fold into lower frequencies, an effect called aliasing. Longer records improve frequency resolution. Faster sampling and an antialiasing filter address a different limitation.

Predict, change, compare, explain

Connect the spectrum to the saved record

Predict how stronger measurement changes the flat readout background. Change only strength and compare shot noise with the mechanical contributions.
Save the observation record, sample interval and spectral convention. Explain why a finite-record spectrum does not coincide point by point with a smooth model spectrum.
Lower efficiency. Distinguish changes in mechanical motion from changes in readout imprecision.

Exercises with explanations

A peak doubles in height and halves in width. Has temperature necessarily increased?

Read the explanation

No. For a high-Q Lorentzian, area scales with peak height times linewidth. The area and mechanical variance can stay fixed.

What happens if you integrate the positive half of a two-sided per-hertz spectrum as though it were one-sided?

Read the explanation

You obtain half the variance because positive and negative frequencies contribute equally for real displacement. The inferred standard deviation is too small by the square root of two.

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.

Callen & Welton, Irreversibility and Generalized Noise, Physical Review 83, 34 (1951)

Foundational relation between linear dissipation and generalized-force fluctuations. The publisher abstract was checked; its full text is subscription-restricted. The course derives its oscillator normalization using the openly available spectral treatment in Aspelmeyer II.2.3.