Enhanced Quantum Nonlinearities in a Two-Mode Optomechanical System

In cavity optomechanics, nanomechanical motion couples to a localized optical mode. The regime of single-photon strong coupling is reached when the optical shift induced by a single phonon becomes comparable to the cavity linewidth. We consider a setup in this regime comprising two optical modes and one mechanical mode. For mechanical frequencies nearly resonant to the optical level splitting, we find the photon-phonon and the photon-photon interactions to be significantly enhanced. In addition to dispersive phonon detection in a novel regime, this offers the prospect of optomechanical photon measurement. We study these quantum nondemolition detection processes using both analytical and numerical approaches.

Figure 1

(a) and (b) Example implementations of the double cavity setup for enhanced quantum nonlinearities: membrane in the middle (a) and optomechanical crystal setup (b). (c) Scheme depicting the mechanical mode $b$ and the optical modes $a_{\pm }$. For the photon and phonon detection applications discussed in this Letter, the cavities are assumed to be driven by independent laser sources and the transmitted signal is measured by photodetectors $D_{\pm }$.

Figure 2

(a) Optical resonances vs mechanical displacement. For $\delta \Omega =2J-\Omega \ll \Omega$, $J$, the regime of enhanced quantum nonlinearity is reached. (b) Energy level scheme of the double cavity optomechanical system. The most relevant second-order transition process is indicated.

Figure 3

Phonon detection in the weak dispersive coupling regime $\delta \omega =g_0^2/\delta \Omega <{\kappa }_{+}$, for single-photon strong coupling $g_0/{\kappa }_{+}=3$: (a) Schematic illustration of the resonances of the detection mode corresponding to phonon number states 0,1,2,3,4. (b) and (d) Quantum trajectories of the photon number in the detection mode, ${\overline{n}}_{+}=⟨a_{+}^{†}{a}_{+}⟩$ (b), and the phonon number, ${\overline{n}}_b=⟨b^{†}b⟩$ (d) from a numerical simulation of the stochastic master equation. $\delta \Omega =20{\kappa }_{+}$, $n_{\mathrm{th}}=2$, $\Gamma ={10}^{-3}{\kappa }_{+}$, ${\alpha }_{+}={\kappa }_{+}$, ${\omega }_{L+}={\omega }_{+}$, ${\kappa }_{-}={10}^{-2}{\kappa }_{+}$, ${\kappa }_{\pm ,t}=0.9{\kappa }_{\pm }$. (c) Photon counts recorded at the photodetector $D_{+}$ within an interval $[t-{\tau }_{\mathrm{meas}},t]$ with ${\tau }_{\mathrm{meas}}=50{\kappa }_{+}^{-1}$.

Figure 4

(a) Spectrum of the detection mode, $⟨a_{-}{a}_{-}^{†}{⟩}_{\omega }=\int e^{i\omega \tau }⟨a_{-}(t+\tau )a_{-}(t{)}^{†}⟩d\tau$, in the presence of a strongly driven signal mode, ${\overline{n}}_{+}=1$. With increasing optomechanical coupling rate $g_0$, the splitting between the resonance peaks grows like $\delta \omega =g_0^2/\delta \Omega$. The inset shows the spectrum for $g_0=20{\kappa }_{-}$ (cut indicated in main figure). (b) and (c) Quantum trajectories for the detection mode driven at (b) the zero-photon resonance, ${\omega }_{L-}-{\omega }_{-}=\delta \omega$, and (c) at the one-photon resonance, ${\omega }_{L-}-{\omega }_{-}=2\delta \omega$, showing anti-correlation and correlation between signal and detection modes. $g_0=20{\kappa }_{-}$, $\delta \Omega =100{\kappa }_{-}$, ${\kappa }_{+}={10}^{-2}{\kappa }_{-}$, ${\kappa }_{\pm ,t}=0.9{\kappa }_{\pm }$, ${\omega }_{L+}={\omega }_{+}$, ${\alpha }_{L-}={\kappa }_{-}/4$, $n_{\mathrm{th}}=0$, $\Gamma ={\kappa }_{-}$.