Backaction limits on self-sustained optomechanical oscillations

The maximum amplitude of mechanical oscillators coupled to optical cavities is studied both analytically and numerically. The optical backaction on the resonator enables self-sustained oscillations whose limit cycle is set by the dynamic range of the cavity. The maximum attainable amplitude and the phonon generation quantum efficiency of the backaction process are studied for both unresolved and resolved cavities. Quantum efficiencies far exceeding one are found in the resolved sideband regime where the amplitude is low. On the other hand, the maximum amplitude is found in the unresolved system. Finally, the role of mechanical nonlinearities is addressed.

Figure 1

(a) Schematic representation of a generic optomechanical system. Laser light with power $P$ enters the Fabry-Pérot cavity through the left mirror and bounces back and forth between the two mirrors. The right mirror acts as a mechanical resonator. (b) Optical backaction in the small amplitude limit. A delay between the displacement $\tilde{u}$ and the change in the photon occupation $\tilde{n}$ due to the finite cavity lifetime ${\tilde{\kappa }}^{-1}$ leads to ellipsoidal trajectories whose area is proportional to the work done on the resonator. Negative (positive) work leads to cooling (heating) of the resonator in the red- (blue-) detuned region. (c) The optical backaction in the large amplitude regime where the oscillator sweeps across the entire cavity resonance. The black line in (b),(c) indicates the position and linewidth of the cavity. (d) Simulated ring up of a mechanical resonator with ${\tilde{\gamma }}_0=0.001$, coupled to a cavity with $\tilde{\kappa }=100$, ${\tilde{\Delta }}_0=30$, and $c_{\mathrm{OM}}=2000$. From this time trace the amplification rate ${\tilde{A}}^{-1}d\tilde{A}/d\tilde{t}$ and oscillation frequency ${\tilde{\omega }}_R$ are derived (e).

Figure 2

Magnitude (a),(b) and phase (c),(d) of the steady-state response of the cavity photon number $n_{\omega }$ to a harmonic displacement with varying amplitude $\tilde{A}$ for ${\tilde{\Delta }}_0+\langle \tilde{u}\rangle =0.30\tilde{\kappa }$. The curves with different values for $\tilde{\kappa }$ are extracted from numerical time-domain simulations (see the Appendix). The right panels show zooms of the area around $\tilde{A}=10$.

Figure 3

Limit cycles of large amplitude motion in the USR for ${\tilde{\gamma }}_0=0.01,$ $\tilde{\kappa }=100,{\tilde{\Delta }}_0=30$, and $c_{\mathrm{OM}}=2\times {10}^5$. (a) Displacement-velocity phase portrait with the two amplitudes ($A_1$ and $A_2$) and the direction of motion indicated. (b) Three-dimensional phase portrait with the photon number on the vertical axis. The black line shows the projection onto the $\tilde{u}$-$\dot{\stackrel{̃}{u}}$ plane. (c) Time trace of the cavity occupation.

Figure 4

Amplitude (a) and quantum efficiency (b) for ${\tilde{\gamma }}_0=0.01$ and ${\tilde{\Delta }}_0=3$, obtained by direct numerical integration of Eqs. (3) and (4) (see the Appendix) The dotted lines in (a) indicate the amplitude predicted by Eq. (11) and the shaded region indicates where $\tilde{A}>\tilde{\kappa }$ and $\tilde{A}<{\tilde{\kappa }}^2$ (see text).

Figure 5

Motion-induced sidebands of the optical field for small (top) and large amplitude (bottom) in the RSR (left) and USR (right). Negative $m$ corresponds to the emission of phonons; positive $m$ corresponds to absorption. For ${\tilde{\Delta }}_0=+1$ the cavity resonance is located at $m=-1$ (dotted lines).

Figure 6

Oscillation frequency (a),(b) and amplitude (c),(d) of nonlinear resonators in the USR ($\tilde{\kappa }=100$, left) and RSR ($\tilde{\kappa }=0.01$, right) plotted against the nonlinearity parameter $\tilde{\alpha }$. Both simulations have been done for ${\tilde{\gamma }}_0={10}^{-5}$, ${\tilde{\Delta }}_0+\langle \tilde{u}\rangle =1$, and $c_{\mathrm{OM}}=1$. The insets show the shape of the mechanical potential energy. For negative $\tilde{\alpha }$ the spring constant weakens (left), and for $\tilde{\alpha }>0$ it stiffens (right).