Amplitude Noise Suppression in Cavity-Driven Oscillations of a Mechanical Resonator

We analyze the amplitude and phase noise of limit-cycle oscillations in a mechanical resonator coupled parametrically to an optical cavity driven above its resonant frequency. At a given temperature the limit-cycle oscillations have lower amplitude noise than states of the same average amplitude excited by a pure harmonic drive; for sufficiently low thermal noise a sub-Poissonian resonator state can be produced. We also calculate the linewidth narrowing that occurs in the limit-cycle states and show that while the minimum is set by direct phase diffusion, diffusion due to the optical spring effect can dominate if the cavity is not driven exactly at a sideband resonance.

Figure 1

Resonator Fano factor $F$ and average energy $⟨n⟩$ (inset) as a function of $\Delta$ calculated using $P(B)$ (dashed red curves) and numerically (solid blue curves). In (a) ${\omega }_m=1$, $g=0.4$, ${\gamma }_m=5\times {10}^{-5}$, and in (b) ${\omega }_m=5$, $g=1.5$, ${\gamma }_m=3\times {10}^{-5}$, in each case $\overline{n}=0$ and $\Omega =0.05$ (we adopt units such that ${\gamma }_c=1$). The results obtained from the Gaussian approximation to $P(B)$ (see text) are shown in (b) (light green dashed curves).

Figure 2

Analytic [Eq. (12)] (dashed red curves) and numerical (solid blue curves) calculations of the resonator Fano factor $F$ in terms of $F_d$. The curves are (top to bottom) $\overline{n}=0,0.5,1,5$. In each case ${\omega }_m=5$, ${\gamma }_m=3\times {10}^{-5}$, $g=2$, and $\Omega =0.05$. The dots represent the on-resonance value calculated using Eq. (13). Note, for $\overline{n}=0$, $F_d=1$ and $F$ reaches a minimum $\simeq 0.9$.

Figure 3

(a) Phase diffusion calculated numerically (solid blue line) and analytically (dashed red line). The solid thin line and the dashed thin line show the direct phase diffusion $(D_m+D_{BA}^{+})/2$ and that due to the optical spring effect $2\delta {\omega }_L^2/{\gamma }_L^2(D_m+D_{BA}^{-})$, respectively. The insets show the linewidths ${\Lambda }_0$ (b) and ${\Lambda }_{{\omega }_m}$ (c). Parameters are ${\omega }_m=5$, ${\gamma }_m=3\times {10}^{-5}$, $g=1.5$, $\Omega =0.05$, $\overline{n}=0$.