How a moving mirror changes light

The cavity length selects the light

Fix one mirror and mount the other on an elastic support. Light travels along the line joining their centres. Define an outward displacement as positive. A longer cavity fits the same number of half wavelengths only if each wavelength increases, so the resonant frequency falls.

Let $L$ be the equilibrium length, $x$ the displacement, $j$ an integer longitudinal mode number and $c$ the vacuum speed of light. The angular frequency $\omega_c$ is measured in radians per second; it is $2\pi$ times ordinary frequency. Follow one optical mode with displacement much smaller than the cavity length.

\omega_c(x)=\frac{j\pi c}{L+x}\simeq\omega_c(0)-Gx,\qquad G=\frac{\omega_c(0)}L>0

Keep the term linear in displacement. The positive pull magnitude times displacement gives the decrease in cavity frequency.

L

L+x

Original schematic. Moving the right mirror outward lengthens the standing wave and lowers resonance. Displacement is magnified.

Source locations: Aspelmeyer

Obtain the force from photon energy

Each photon has energy $\hbar\omega_c$ . With $n$ photons stored, take the negative displacement derivative of the optical energy. It gives a force in the positive direction. As the mirror moves outward, light loses energy that can enter mechanical motion.

This is the energy description of the momentum transferred at reflection. It ties the optical frequency shift and mechanical force to one interaction. Photon-number fluctuations therefore produce force fluctuations. The measurement tradeoff developed later already has a physical origin here.

H_{\rm int}=-\hbar Gxn,\qquad F_{\rm rad}=\hbar Gn

The derivative holds photon number fixed. A continuously driven cavity also needs an equation for photons entering and escaping.

Source locations: Marquardt et al.

Choose a length for mechanical fluctuations

A mirror can vibrate in many shapes. Choose one mode, with effective mass $M$ and mechanical angular frequency $\Omega_m$ . Effective mass belongs to the specified displacement coordinate. It need not equal the mass of the whole support. Define $x_{\rm zpf}$ as the ground-state position standard deviation and $g_0$ as the cavity frequency pull over that distance.

Single-photon coupling is a device property. Adding light can enhance an interaction around a driven operating point, but it does not change the effect assigned to one photon in the same device.

Worked example: take $L=1\,\mathrm{mm}$ , $\lambda=1064\,\mathrm{nm}$ , $M=10^{-12}\,\mathrm{kg}$ and $\Omega_m/(2\pi)=1\,\mathrm{MHz}$ . First obtain $x_{\rm zpf}=2.90\,\mathrm{fm}$ . Then $G=2\pi c/(\lambda L)$ gives $g_0/(2\pi)=817\,\mathrm{Hz}$ . Whether that shift is resolved depends on the cavity linewidth.

x_{\rm zpf}=\sqrt{\frac{\hbar}{2M\Omega_m}},\qquad g_0=Gx_{\rm zpf},\qquad x=x_{\rm zpf}(b+b^\dagger)

The last expression introduces lowering and raising operators for a mechanical quantum. Their algebra will be used later.

Predict, change, compare, explain

Follow the two directions of the interaction

Predict how outward mirror displacement shifts resonance. Then drag the gold laser dial to change laser frequency relative to resonance and compare the readout.
Hold linewidth fixed and compare red- and blue-detuned demonstrations. Describe separately how motion changes light and how light changes motion.
Use the femtometre scale in the example to explain the schematic magnification of the displayed motion.

Exercises with explanations

Increase mass fourfold at fixed mechanical frequency and cavity length. What happens to zero-point displacement and single-photon coupling?
Read the explanation
Both halve: zero-point displacement scales with inverse square root of mass, and the optical pull per unit displacement stays fixed.
Does one hundred times more intracavity light increase single-photon coupling one hundredfold?
Read the explanation
Single-photon coupling stays fixed. Mean radiation pressure increases one hundredfold. The enhanced linearized coupling introduced later increases tenfold because it scales with the square root of photon number.

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.
Marquardt et al., Quantum theory of cavity-assisted sideband cooling of mechanical motion, arXiv:cond-mat/0701416v1 (2007)
Weak-coupling force-noise transition rates and the resolved-sideband cooling limit. The browser uses the single-mode, weak-coupling rate model; strong-coupling normal-mode splitting needs the full coupled response.