Parametric Normal-Mode Splitting in Cavity Optomechanics

Recent experimental progress in cavity optomechanics has allowed cooling of mesoscopic mechanical oscillators via dynamic backaction provided by the parametric coupling to either an optical or an electrical resonator. Here we analyze the occurrence of normal-mode splitting in backaction cooling at high input power. We find that a hybridization of the oscillator’s motion with the fluctuations of the driving field occurs and leads to a splitting of the mechanical and optical fluctuation spectra. Moreover, we find that cooling experiences a classical limitation through the cavity lifetime.

Figure 1

(a) Electromechanical realization of parametric coupling of a mechanical oscillator to an $LC$ circuit, where the coupling is determined by $\frac{d{\omega }_p}{dx}{|}_{x=0}=\frac{{\omega }_p{C}_c}{2d{C}_{\mathrm{tot}}}{|}_{x=0}$ ($C_{\mathrm{tot}}$ is the total capacitance). (b) Optomechanical realization of parametric coupling of a mechanical oscillator to a Fabry-Perot optical mode with $\frac{d{\omega }_p}{dx}{|}_{x=0}=-\frac{{\omega }_p}{L}{|}_{x=0}$ ($L$ is the cavity length).

Figure 2

Real and imaginary parts of the eigenvalues [cf. Eqs. (3, 4)] of the linearized cooling problem corresponding, respectively, to the eigenfrequency and mode damping for $\Delta =-{\Omega }_m$ and $\kappa /{\Omega }_m=0.2$. The inset magnifies the resonance shift before the mode splitting. The real part is underlaid with the normalized classical displacement spectrum (contribution $\propto n_m$) [cf. Eq. (2)]; thereof sample curves are highlighted in Fig. 3 for the rates marked by the dots.

Figure 3

(a) Normalized logarithm of the classical displacement spectrum (contribution $\propto n_m$) [cf. Eq. (2)] as a function of the normalized detuning for $g_m/{\Omega }_m=0.4$ and $\kappa /{\Omega }_m=0.2$. (b) Typical displacement spectra for coupling rates that are represented by the dots in Fig. 2. The solid curves above correspond to the phase spectral density $S_{\Phi }(\omega )$ measured in homodyne detection.