Why cooling needs a cavity that stores light

Calculate the chances of removing and adding phonons

Use enhanced coupling $g$ , energy linewidth $\kappa$ , mechanical frequency $\Omega_m$ and effective laser-minus-cavity detuning $\Delta$ . In weak coupling, cavity response settles faster than mechanical energy changes. Its response at the two sidebands gives cooling coefficient $A_-$ and heating coefficient $A_+$ .

With $n$ phonons present, downward transitions occur at $nA_-$ and upward transitions at $(n+1)A_+$ . The extra one allows Stokes scattering to excite the ground state. It supplies a quantum source of residual motion.

A_- = \frac{g^2\kappa}{(\kappa/2)^2+(\Delta+\Omega_m)^2},\qquad A_+ = \frac{g^2\kappa}{(\kappa/2)^2+(\Delta-\Omega_m)^2}

Rates have units of inverse seconds. This single-mode rate treatment assumes weak coupling and a stable operating point.

Source locations: Marquardt et al. · Aspelmeyer

Keep the original thermal bath

The mechanical mode also exchanges energy with its environment at rate $\Gamma_m$ . A bath at temperature $T$ has equilibrium occupation $n_{\rm th}$ . Add the phonon changes from each transition. Optical damping is the difference between cooling and heating coefficients. The steady occupation is total heating divided by total damping.

Stronger optical damping reduces the original bath contribution. Stokes events still set the optical bath temperature. If the cavity is broad, both sidebands pass and phonon creation becomes difficult to suppress.

\begin{aligned}\dot{\bar n}&=-(\Gamma_m+A_--A_+)\bar n+\Gamma_m n_{\rm th}+A_+,\\ n_{\rm ss}&=\frac{\Gamma_m n_{\rm th}+A_+}{\Gamma_m+A_--A_+},\qquad n_{\rm th}=\frac1{e^{\hbar\Omega_m/(k_BT)}-1}.\end{aligned}

A reachable linear steady state requires positive total damping. Stokes heating remains even at zero bath temperature.

Work an example before taking the ideal limit

Normalize rates by $\Omega_m$ . Choose $\Delta/\Omega_m=-1$ , $\kappa/\Omega_m=0.2$ , $g/\Omega_m=0.03$ , $\Gamma_m/\Omega_m=0.001$ and $n_{\rm th}=10$ . Then $A_-=0.018\Omega_m$ and $A_+=4.49\times10^{-5}\Omega_m$ .

Total damping is $0.018955\Omega_m$ . Divide heating $0.0100449\Omega_m$ by it to obtain $n_{\rm ss}=0.530$ . The optical bath alone would have occupation $0.0025$ . Thermal leakage accounts for most of the difference.

More drive can reduce that leakage fraction, but enhanced coupling must remain small enough for the rate model. When coherent exchange between optical and mechanical modes becomes important, solve their coupled dynamics.

n_{\rm opt}=\frac{A_+}{A_--A_+}=\left(\frac{\kappa}{4\Omega_m}\right)^2\quad(\Delta=-\Omega_m)

This is the optical-bath occupation at the specified red-sideband detuning, separate from the final occupation with a finite mechanical bath.

Follow blue driving to the threshold

Blue-sideband driving favours upward transitions. Its negative damping first cancels ordinary mechanical loss. Beyond that point, a small vibration grows. The linear model predicts a growth rate.

A real device may saturate into self-oscillation, requiring nonlinear cavity response and device-specific losses. Clipping a growing amplitude to the display boundary would not model that saturation. At threshold the small-signal interpretation needs to be explicit.

\Gamma_{\rm eff}=\Gamma_m+A_--A_+>0

Positive total damping gives decay. Zero marks the rate-model threshold; negative total damping gives growth of a small disturbance.

Predict, change, compare, explain

Find what limits cooling first

Predict whether a broad or narrow cavity more readily approaches the ground state. Compare the presets and scattering rates.
Increase drive at fixed red detuning. Follow thermal and backaction contributions and read the weak-coupling notice.
Approach the blue-detuned threshold from low drive. Use total damping to explain decay or growth at this operating point.

Exercises with explanations

The steady-state formula gives a negative occupation when damping is negative. What does this mean?

Read the explanation

The linear steady state is unstable. The dynamical solution grows instead of approaching that algebraic fixed point. Large-amplitude motion requires nonlinear dynamics.

Halve linewidth at fixed red-sideband detuning. Does final occupation always fall fourfold?

Read the explanation

The specified optical-bath limit falls fourfold. Actual occupation also depends on cooling rate, mechanical damping and thermal occupation. Dominant thermal leakage prevents this simple scaling.

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.