Fig. 1. Schematic diagram of the optically coupled PT optomechanical system. (a) μC1 denotes an active WGM resonator with gain medium, and μC2 is a passive one. (b) Equivalent diagram of the PT optomechanical system, where the WGM resonators are replaced by Fabry–Perot cavities with a movable end mirror and a fixed one. The two cavities are directly coupled through the inter-cavity evanescent optical fields, and the optical coupling strength κ depends on the distance between the two Fabry–Perot cavities.

Fig. 2. (a) Mode splitting of the supermodes, i.e., the real parts of the eigenfrequencies and (b) linewidth of the supermodes, i.e., the imaginary parts of the eigenfrequencies. The green region is the broken-PT-symmetry regime, and the pink region corresponds to the PT-symmetry regime.

Fig. 3. (a) Optomechanics-induced mechanical frequency shifts δΩ1,2 of the two optomechanical resonators versus the optical coupling strength κ both in the broken-PT-symmetric regime and the PT-symmetric regime. (b) Effective coupling strength κmech between two mechanical modes versus the optical coupling strength κ.

Fig. 4. (a) Effective mechanical frequencies Ω1,eff and Ω2,eff versus the optical coupling strength κ, where the red solid (blue dashed) curve represents the frequency of β1 (β2) and the light green (pink) area is the broken-PT-symmetric (PT-symmetric) regime. (b) Numerical results of cross-correlation Mcc with different values of κ in broken-PT-symmetric and PT-symmetric regimes. (c) Spectrograms of mechanical modes x1 and x2 with increasing optical coupling strength κ in the broken-PT-symmetric regime, where the nature frequencies Ω1,2 of x1,2 are 5 MHz and 15 MHz, respectively. Here, κ↑ (κ↓) denotes the increasing (decreasing) of κ from 2 MHz and 29.86 MHz (50 MHz to 30.81 MHz), and the left and right red arrows indicate the moving direction of the spectra of x1 and x2 by increasing (decreasing) κ, as shown in (c) [(d)]. (d) Spectrograms of mechanical modes x1 and x2 with decreasing optical coupling strength κ in the PT-symmetric regime, in which weaker coupling strength κ makes the two resonators more easily be synchronized.

Fig. 5. (a) Effects of the stochastic noises on Mcc with respect to different stochastic noise intensity D in the broken-PT-symmetric regime with κ=27.76  MHz. (b) Variances of Mcc versus noise level D in (a). (c) Effects of the stochastic noises on Mcc versus different D in the PT-symmetric regime with κ=32.19  MHz. The variance of Mcc is presented in (d).

Fig. 6. Kramers rates r1 and r2 of mechanical displacements x1 and x2 versus the noise intensity D in broken-PT-symmetric and PT-symmetric regimes. (a) The red curve (blue curve) represents the curve for Kramers rate r1 (r2) versus the noise intensity D in the broken-PT-symmetric regime. Here, the optical coupling strength κ=27.76  MHz is fixed. (b) The simulation results of Kramers rates r1 and r2 versus noise intensity D in the PT-symmetric regime, where the optical coupling strength is fixed as κ=32.19  MHz.

Fig. 7. Numerical results of the normalized correlation function R with different values of temperature T in broken-PT-symmetric and PT-symmetric regimes, where Tr denotes the room temperature. (a) Effects of the thermal noises on R with respect to different temperature T in the broken-PT-symmetric regime with κ=27.76  MHz. (b) Effects of the thermal noises on R versus T in the PT-symmetric regime with κ=32.19  MHz.

Fig. 8. Kramers rates r1 and r2 of the mechanical displacements x1 and x2 versus the temperature T in both broken-PT-symmetric and PT-symmetric regimes, where Tr is the room temperature. (a) The red curve (blue curve) denotes the Kramers rate r1 (r2) with increasing temperature T in the broken-PT-symmetric regime, where the optical coupling strength κ=27.76  MHz is fixed. (b) The Kramers rates r1 and r2 versus the temperature T correspond to the PT-symmetric regime (κ=32.19  MHz).

Fig. 9. (a) Optomechanics-induced mechanical frequency shifts δΩ1,2 versus the optical coupling strength κ in the broken-PT-symmetric regime (light green area) and PT-symmetric regime (pink area). Here, we fix Δ2=5  MHz and plot the curves of δΩ1,2 for different Δ1. The solid (dashed) curves denote the curves of the mechanical frequency shift δΩ1 (δΩ2) with different Δ1. (b) The effective mechanical coupling strength κmech between the two mechanical modes versus the optical coupling strength κ.

Fig. 10. Effective optomechanical coupling strength geff versus the optical coupling strength κ. In the green area, the system is far away from the EP, and the effective optomechanical coupling strength geff is linearly dependent on κ. In the pink area, the system is in the vicinity of the EP, and, in this case, geff changes nonlinearly with κ.

Fig. 11. (a) Optomechanics-induced mechanical frequency shifts δΩ1 versus the optical coupling strength κ with different Γ−. The solid curve denotes the case where gain and loss are balanced, i.e., Γ−=0. It is shown that the amplification effects of δΩ1 are suppressed with the increase of Γ−. (b) Corresponding optomechanics-induced mechanical frequency shifts −δΩ2 versus the optical coupling strength κ with different Γ−. (c) Effective mechanical coupling κmech between the two mechanical modes versus the optical coupling strength κ with different Γ−. It is shown that the amplification of κmech is also suppressed with the increase of Γ−.