How strong light enhances a weak interaction

Find the average intracavity field

Let the laser frequency be $\omega_L$ and define $\Delta=\omega_L-\omega_c$ relative to the cavity including its static displacement. Red detuning is negative. Cavity energy decays at $\kappa$, so amplitude decays at $\kappa/2$. Input coupling contributes $\kappa_{\rm ex}$; $|s_{\rm in}|^2$ is incident photon flux.

In a frame rotating with the laser, a steady drive stops rotating rapidly. Escape, detuning and replenishment set its mean amplitude. Find this operating point before calculating the effect of small motion.

Source locations: Aspelmeyer

Retain the first fluctuation terms

Write $a=\alpha+d$ and measure motion from the displaced mechanical equilibrium. Choose drive phase so $\alpha$ is real and positive. Expanding photon number gives a constant, a term proportional to $d+d^\dagger$, and $d^\dagger d$. When fluctuations are small relative to the mean field, remove the mean force and neglect the last interaction term.

The remaining interaction couples one optical fluctuation to displacement. Linear equations preserve a Gaussian state, whose phase-space shape is specified by its centre and covariance. This will make continuous state estimation tractable.

Ask separately whether this working point is stable and whether the omitted interaction is small. Stability prevents unbounded growth of a perturbation. Accuracy also depends on fluctuations, the background, linewidth and the observable being calculated.

Follow energy through the two scattering channels

Motion at $\Omega_m$ creates optical sidebands on either side of the carrier. An outgoing photon that gains one mechanical quantum removes a phonon: the anti-Stokes process. A photon that loses that energy creates a phonon: the Stokes process. A phonon is one energy quantum of the chosen mechanical mode.

With the laser one mechanical frequency below cavity resonance, the upper anti-Stokes sideband is resonant. The lower Stokes sideband lies farther away. A narrow cavity therefore favours removal of mechanical energy. Blue detuning reverses the preference.

Worked example: let $g_0/(2\pi)=100\,\mathrm{Hz}$ and $\bar n_c=10^6$. Then $g/(2\pi)=100\,\mathrm{kHz}$. If $\kappa/(2\pi)=1\,\mathrm{MHz}$, enhanced coupling is still below linewidth. The single-photon ratio remains $g_0/\kappa=10^{-4}$. A bright drive has enhanced a linear interaction without creating a large single-photon nonlinearity.

Source locations: Marquardt et al.

Locate the limits of the expansion

Large displacement can sweep the cavity through a substantial part of its linewidth, invalidating a local-slope expansion. Optical fluctuations comparable to the background restore the discarded interaction. Near instability, perturbations grow until a fixed-state linearization ceases to describe them.

A separate route makes the effect of one photon large. Then adding the first and second photons costs different energies even under weak drive. The final chapter keeps the full photon-number–displacement interaction to study that photon blockade.