Fig. 1. (a), (b) Schematic diagram for generating QOM coupling, where a mechanical nanostring oscillator is placed between two whispering gallery mode (WGM) resonators. (c), (d) Dispersion of the optical modes as a function of the displacement.

Fig. 2. (a) The transmission coefficients T21 (solid black curve) and T12 (dashed red curve) as a function of the detuning Δ/G. (b) The isolation as a function of the detuning Δ/G. (c) The equal-time second-order correlation function log10[gij(2)(0)] (ij=12,21) as a function of the detuning Δ/G. (d) The second-order correlation function log10[g21(2)(τ)] as a function of the normalized time delay γcτ/(2π) at detuning Δ=2G. The other parameters are Δm=Δ/2, G=3γc, ε=γc/10, γm=γc/100, and nth=0.

Fig. 3. Schematic energy spectrum of the linearized QOM coupling between optical mode aL and mechanical resonator b, where |00⟩≡|0,0⟩, |10⟩≡|0,1⟩, |2±1⟩≡(|1,0⟩±|0,2⟩)/2, |3±1⟩≡(|1,1⟩±|0,3⟩)/2, |40⟩≡(−3|2,0⟩+|0,4⟩)/2, |4±1⟩≡(|2,0⟩±2|1,2⟩+3|0,4⟩)/(22), and |n,m⟩ represents the Fock state with n photons in aL and m phonons in b.

Fig. 4. (a) Transmission coefficient T21. (b) The equal-time second-order correlation function log10[g21(2)(0)] versus the detuning Δ/G with different mean thermal phonon number (nth=0,0.1,1). (c) The isolation T21/T12. (d) The equal-time second-order correlation function log10[g21(2)(0)] versus the mean thermal phonon number nth with different detuning (Δ=0,2G,6G). The other parameters are the same as in Fig. 2.