Abstract

Synchronization has great impacts in various fields such as self-clocking, communication, and neural networks. Here, we present a mechanism of synchronization for two mechanical modes in two coupled optomechanical resonators with a parity-time (PT)-symmetric structure. It is shown that the degree of synchronization between the two far-off-resonant mechanical modes can be increased by decreasing the coupling strength between the two optomechanical resonators due to the large amplification of optomechanical interaction near the exceptional point. Additionally, when we consider the stochastic noises in the optomechanical resonators by working near the exceptional point, we find that more noises can enhance the degree of synchronization of the system under a particular parameter regime. Our results open up a new dimension of research for PT-symmetric systems and synchronization.

© 2021 Chinese Laser Press

1. INTRODUCTION

Synchronization is a phenomenon in which two or more systems coordinate and act in the same time with similar behaviors. It can be extensively observed in our daily life, such as the chorusing of crickets, flash of fireflies, pendulum clocks, firing neurons, applauding of audiences, and even the life cycle of creatures [13]. As synchronization is a qualitative transition where the rhythms of two or more different objects are adjusted in unison, it also attracts great interest and is widely applied to various fields, such as data communication, time keeping, navigation, cryptography, and neuroscience [49].

Due to the recent developments of nano-fabrication techniques, especially those for high-quality-factor on-chip optomechanical resonators [10], it is possible to demonstrate synchronization in on-chip nano-scale platforms [1116]. For example, in Ref. [13], a pair of closely placed optomechanical resonators with different mechanical frequencies was synchronized by indirect coupling through the coupled optical fields. In Ref. [14], two nano-mechanical oscillators separated by about 80 μm were synchronized through the same optical field in an optical racetrack. Moreover, in Ref. [15], master–slave frequency locking was realized between two separated optomechanical oscillators (3.2 km apart) through the light.

Although synchronization in optomechanical resonators has been explored for several years both theoretically and experimentally, there are still several problems left open. One of them is that the frequency mismatch between two synchronized oscillators is required to be very small compared with the inherent frequencies of the two oscillators [1216]. In the existing experiments [13,15], the frequency differences between the two micro resonators are about dozens of kilohertz (kHz, 70 to 80 kHz), which are far smaller than the natural frequencies [about dozens of megahertz (MHz)] of the two oscillators. In Ref. [12], the authors used the unidirectionally cascaded structure to synchronize two resonators and pointed out that the intrinsic frequency mismatch should be limited to around 1% of the mechanical frequency of the first resonator. In fact, the restriction of intrinsic frequency mismatch between two micro resonators mainly stems from the fact that the vacuum optomechanical coupling strength, i.e., the coupling strength between single photon and single phonon, is traditionally very tiny (kHz) [10]. The field-enhanced optomechanical coupling strength is still not strong enough to achieve the strong coupling regime even when it is amplified by the input field [17], which thus results in the limitation of synchronization between micro resonators.

In the past few years, the parity-time (PT)-symmetric systems, e.g.,  coupled resonators composed of an active resonator and a passive one, have attracted great attentions [1827]. In such kind of systems, various interesting properties, such as topological singularity and possible applications, have been studied both theoretically [2834] and experimentally [3543]. Especially, in the PT-symmetric optomechanical systems, in which PT-symmetric optical modes are coupled to mechanical modes, the optomechanical coupling strength can be greatly enhanced due to the topological singularity at the exceptional point (EP) [2932].

Motivated by such merits of the PT-symmetric structure, we study the frequency synchronization in PT-symmetric optomechanical systems. It is shown that the far-off-resonant mechanical oscillations in two coupled PT-symmetric optomechanical resonators can be synchronized. Although the optomechanical interaction in our system will influence the EP of the PT-symmetric structure, this influence is negligibly small under the parameter regime we consider [2934]. Besides, by introducing the PT-symmetric structure, we observe an interesting phenomenon, where the two mechanical modes of the coupled optomechanical resonators tend to oscillate in unison by decreasing the optical coupling strength between them. This observation somewhat conflicts with the normal phenomenon where the stronger the coupling strength between two systems is, the easier the synchronization can be realized. Another counterintuitive phenomenon is observed when we consider the noises acting on the optomechanical resonators. It is shown that more noises may benefit the synchronization process between the two mechanical modes.

This paper is organized as follows. In Section 2, we present the PT-symmetrical optomechanical system we consider. In Section 3, we show how to achieve mechanical frequency synchronization between the two far-off-resonant optomechanical resonators and also analyze the physical mechanism where the frequency synchronization is easier to be realized by decreasing the inter-cavity optical coupling strength. In Section 4, we discuss the positive roles of the optical stochastic noises and the mechanical thermal noises for synchronization, respectively. We summarize our results in Section 5.

2. COUPLED-OPTOMECHANICAL RESONATORS WITH OPTICAL PT-SYMMETRIC STRUCTURE

As shown in Fig. 1(a), the system we consider here consists of two coupled whispering-gallery-mode (WGM) resonators. The left WGM resonator (μC1) is an active one that can be realized, e.g.,  by an Er3+-doped silica disk, and the right one (μC2) is a passive resonator. Each resonator supports an optical mode αi and a mechanical mode βi (i=1,2), and the two resonators are coupled through the inter-cavity evanescent optical fields with coupling strength κ. As is well known, although the two mechanical modes β1 and β2, located in two different resonators, are not directly coupled, they can be indirectly coupled through the inter-cavity optical coupling and the intra-cavity optomechanical coupling. We elaborate this indirect mechanical coupling in Fig. 1(b). Each WGM resonator is equivalent to a Fabry–Perot cavity, with one fixed mirror and one movable one. The optical modes α1 and α2 represent the optical fields in the Fabry–Perot cavities, and the mechanical modes β1 and β2 indicate the motions of the movable mirrors. In each equivalent Fabry–Perot cavity, the movable mirror suffers a radiation-pressure force induced by the optical mode αi (i=1,2). Such a force is proportional to the circulating optical intensity |αi|2 in the cavity, which leads to the mechanical motion βi. In the meantime, the movable mirror induces a frequency shift of the optical mode in the cavity, which influences the dynamics of αi. In Fig. 1(b), α1 (α2) and β1 (β2) interact with each other through this kind of radiation-pressure coupling, and α1 and α2 are directly coupled through the inter-cavity evanescent optical fields and therefore lead to the indirect coupling between mechanical modes β1 and β2. In other words, these two mechanical modes correspond to optical modes in two different WGM cavities, respectively, and are indirectly coupled with each other through the direct inter-cavity coupling between the corresponding two optical modes.

 figure: Fig. 1.

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.

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The PT-symmetric optomechanical system we consider can be represented by the following equations:

α˙k=Γopkαkiκα3kigomαk(βk+βk*)+2γkexϵk,β˙k=(Γmk+iΩk)βkigom|αk|2,
where Γop1=γ1+iΔ1 and Γop2=γ2+iΔ2. γk, γkex, Δk=ωckωL, and ϵk (k=1,2) denote the gain (loss) rate of the resonator μCk, the external damping rate induced by the coupling between the resonator and the input/output fiber taper, the detuning frequency between the resonance frequency (ωck) of the cavity mode and the frequency (ωL) of the driving field, and the amplitude of the driving field, respectively. Ωk and Γmk represent the frequency and damping rate of the mechanical mode βk. gom is the vacuum optomechanical coupling strength. Without loss of generality, here we assume that Ω2Ω1. To simplify our discussion, we assume that the gain cavity μC1 and the lossy cavity μC2 have the same vacuum optomechanical coupling strength gom. We also assume that the gain and loss in the system are well balanced, i.e., γ2=γ1γ. Additionally, we consider the case of critical coupling such that γ1ex=γ2ex=γ/2, under which a phase transition point exists called the EP, corresponding to a critical inter-cavity coupling strength κEP=γ. When κ>κEP, which is in so-called PT-symmetric regime, two non-degenerate optical supermodes exist with the same damping rate. When κκEP, which is in the so-called broken PT-symmetric regime, the two optical supermodes are degenerate but with different damping rates. When the system is far away from the EP, the interaction between the optical supermodes and mechanical modes, i.e., the effective radiation-pressure coupling in the supermode picture, is weak. However, this kind of interaction will be greatly enhanced as κ gets closer to κEP. This comes from the topological-singularity-induced amplification of the optomechanical nonlinearity in the vicinity of the EP [2932].

Slightly different from Refs. [2933], in this work, we assume that the difference between the frequency detunings Δ1 and Δ2 is non-zero, i.e., Δ=Δ2Δ10. This non-zero difference between the frequency detunings Δ can eliminate the degeneracy of the optical supermodes at the EP and thus affect the PT-symmetric structure in our system. Therefore, in this work, we need to emphasize that the difference Δ between the two frequency detunings Δ1,2 should be small, such that the non-degeneracy of the optical supermodes at EP is very small, and the PT-symmetric structure of the optical modes is still maintained. In fact, by eliminating the mechanical modes in Eq. (1), the dynamics of the two optical modes can be given by

ddt[α1α2]=i[2Δ¯1+iγκκ2Δ¯2iγ][α1α2]+[2γ1exϵ12γ2exϵ2],
where Δ¯k=(Δk+Δks)/2, and Δ1s and Δ2s are the optical frequency shifts induced by the optomechanical interaction (for details see Appendix A). By diagonalizing the coefficient matrix in Eq. (2), we can obtain two optical supermodes α±, where the corresponding eigenfrequencies can be given by
ωo+=Δ¯1+Δ¯2±i[γ+i(Δ¯2Δ¯1)]2κ2.

Since Δks is very small compared with the frequency detuning Δk, the eigenfrequencies of the two optical supermodes can be reduced to

ωo±=Δ1+Δ22±i(γ+iΔ2)2κ2.

It can be shown in Eq. (4) that both optical supermodes will be degenerate at the EP (κ=γ) if Δ=0. However, if Δ is non-zero, the degeneracy between these two supermodes will be eliminated.

Then, let us assume that

gomΔ<Δ1,2γ,κ,
and the two optical driving fields have the same amplitude, i.e., ϵ1=ϵ2=ϵ. In this case, this non-degeneracy between the two supermodes at the point κ=γ can be evaluated as follows:
Δsplitγ2|Δ+4gom2Ω2Ω1Ω1Ω21Δ2ϵ2|1γ,
where Δsplit=Re[ωo+ωo]Im[ωo+ωo]. The non-degeneracy Δsplit described in Eq. (6) can be simplified further as Δsplit/γ3Δ/γ under the condition gom/Δγ,κ, which means that this non-degeneracy is negligibly small when Δ is small enough. Given the system parameters γ=30MHz, Δ1=4.2MHz, Δ2=5MHz, Ω1=5MHz, Ω2=15MHz, Γm1=8kHz, Γm2=8kHz, gom=3kHz, and ϵ=70MHz1/2, the simulation results of the mode splitting and linewidth of the optical supermodes are shown in Figs. 2(a) and 2(b), respectively. The simulation results of the non-degeneracy Δsplit at the EP in Fig. 2 are around 6 MHz. It is obvious that this non-degeneracy Δsplit at the EP is negligibly small compared with the optical damping rate γ, and the broken-PT-symmetric and PT-symmetric regimes can be clearly observed, as illustrated in Fig. 2. It should be noted that although one eigenfrequency of the optical supermodes has a positive imaginary part, which corresponds to a positive damping rate in the broken-PT-symmetric regime [Fig. 2(b)], the saturation nonlinearity induced by the optomechanical coupling will suppress the divergence induced by this positive damping rate [44,45]. It also should be noted that the non-zero difference between the frequency detunings Δ can break the PT-symmetric optical structure by lifting the degeneracy between the optical supermodes. However, as Δ is so small, i.e., ΔΔ1,2, where the non-degeneracy between optical supermodes is negligibly small, the properties of the PT-symmetric structure, including the mode-splitting of the supermodes [Fig. 2(a)] and the great amplification of optomechanical interaction mentioned in the later discussion, are maintained. Thus, we still name our system in this paper as a PT-symmetric optomechanical system in accord with previous literatures [2933].
 figure: Fig. 2.

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.

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3. FREQUENCY SYNCHRONIZATION VIA PT-SYMMETRY

In order to elaborate how the enhancement of the optomechanical interaction in the vicinity of the EP affects the synchronization between the two mechanical modes, we adiabatically eliminate the degrees of freedom of the optical modes. This enhanced optomechanical coupling will induce significant optomechanics-induced frequency shifts δΩ1 and δΩ2 for the mechanical modes β1 and β2. In fact, under the condition that gomΔ<Δ1,2γ,κ, and ϵ1=ϵ2ϵ, δΩ1 and δΩ2 near the EP can be given by (for detailed derivation see Appendix B)

δΩ1=δΩ2gom2Δ(γ2+κ2)2γϵ2[(κ2γ2)2+γ2Δ2]2.

We show in Fig. 3(a) the optomechanics-induced mechanical frequency shifts δΩ1 (red-solid curve) and δΩ2 (blue-dashed curve) of the two resonators versus the optical coupling strength κ, both in broken-PT-symmetric and PT-symmetric regimes. When the system is far away from the EP, the optomechanics-induced mechanical frequency shift δΩi is negligibly small. However, δΩi will be greatly enhanced such that δΩi is almost comparable with or even larger than Ωi when κ approaches κEP. As these two enhanced frequency shifts for the mechanical modes are opposite in sign, they will lead to significant modifications of mechanical frequencies Ω1 and Ω2 and make the two mechanical frequencies Ω1 and Ω2 approach each other.

 figure: Fig. 3.

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.

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Moreover, in the vicinity of the EP, this enhanced optomechanical coupling can also induce an enhancement of the effective mechanical coupling strength κmech between the mechanical modes β1 and β2. Similarly, by adiabatically eliminating the degrees of freedom of the optical modes, this effective coupling strength κmech between the two mechanical modes β1 and β2 can be expressed as

κmech4gom2Δκ2γ3ϵ2[(κ2γ2)2+γ2Δ2]2.

In Fig. 3(b), the effective mechanical coupling strength κmech versus the optical coupling strength κ is plotted both in broken-PT-symmetric and PT-symmetric regimes. It can be clearly seen that the effective mechanical coupling strength κmech is very tiny when the system is far away from the EP, but it can be significantly enhanced when κ approaches κEP. This enhanced effective mechanical interaction in the vicinity of the EP can also contribute to the synchronization between the two mechanical modes β1 and β2.

Actually, the effective mechanical frequencies of the two mechanical oscillators can be expressed as Ω1,eff=Ω1+δΩ1+δΩcoup and Ω2,eff=Ω2+δΩ2δΩcoup, respectively, where δΩcoup comes from the effective mechanical coupling strength κmech (see Appendix D). This means that the enhanced optomechanics-induced mechanical frequency shifts δΩ1,δΩ2 and effective mechanical coupling strength κmech can lead to significant modifications of mechanical frequencies Ω1,Ω2 together and thus jointly contribute to the synchronization between the two mechanical oscillators, i.e., Ω1,eff=Ω2,eff. We show in Fig. 4(a) the effective mechanical frequencies Ω1,eff (red-solid curve) and Ω2,eff (blue-dashed curve) of the two resonators versus the optical coupling strength κ, both in broken-PT-symmetric and PT-symmetric regimes. It is clear that the two mechanical oscillators tend to be resonant with each other, i.e., Ω1,eff=Ω2,eff, and thus synchronize when κ approaches κEP. As is well known, the frequency mismatch between two synchronized oscillators should be very small in traditional lossy systems [12,13], i.e., |Ω1Ω2|Ω1,Ω2. However, as shown in Fig. 4, our PT-symmetric system can perfectly synchronize two far-off-resonant mechanical oscillators. Actually, as shown in Fig. 4(a), where the nature frequencies Ω1=5MHz and Ω2=15MHz, the effective mechanical frequencies Ω1,eff and Ω2,eff of the two optomechanical resonators coincide with each other when κ approaches κEP, even when the nature frequency mismatch |Ω1Ω2| (10 MHz) is comparable with the nature frequencies Ω1,2 of the two mechanical oscillators, i.e., |Ω1Ω2|Ω1,2. The oscillators with such large frequency mismatch cannot be synchronized in the traditional lossy systems [12,13].

 figure: Fig. 4.

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.

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In addition, we find a counterintuitive phenomenon, where weaker coupling between two optomechanical resonators may be helpful for synchronization for our PT optomechanical system. In fact, as shown in Fig. 4(a), in the PT-symmetric regime (the pink region), when the coupling strength κ between two resonators is decreased, the effective mechanical frequencies of the two resonators tend to coincide with each other, which means that β1 and β2 are inclined to oscillate in unison with the weaker coupling strength κ in the PT-symmetric regime. The broken-PT-symmetric regime is the normal regime where stronger coupling between the two optomechanical resonators makes the two mechanical modes β1 and β2 be inclined to be synchronized. We can more easily see this phenomenon by plotting the spectra of the normalized mechanical displacements of the two optomechanical resonators x1=(β1+β1*)/2 (the red solid curve) and x2=(β2+β2*)/2 (the blue dashed curve) in Figs. 4(c) and 4(d), where κ is increased from 2 MHz to 29.86 MHz in Fig. 4(c) and is decreased from 50 MHz to 30.81 MHz in Fig. 4(d). In Figs. 4(c) and 4(d), the red solid and blue dashed Lorentz peaks represent the spectra of x1 and x2, respectively. The left and right red arrows, as shown in Fig. 4(c) [Fig. 4(d)], separately denote the moving direction of spectra of x1 and x2 by increasing (decreasing) the optical coupling strength κ. The left red solid and right dashed peaks are located at the nature frequencies of x1 (5 MHz) and x2 (15 MHz) when κ (2 MHz) is far away from the EP κEP (30MHz), and approach each other and overlap in the middle position at 10.51 MHz while κ is increased to 29.86 MHz, as shown in Fig. 4(c). In other words, with the increase of the inter-cavity coupling strength κ in the broken-PT-symmetric regime, the Lorentz spectra of x1 and x2 get closer and finally coincide with each other, which means that the two mechanical modes are resonant. However, contrary to our intuition, the resonant peaks of x1 and x2 get closer by decreasing κ in the PT-symmetric regime, as shown in Fig. 2(d). This counterintuitive phenomenon stems from the fact that synchronization only takes place when we approach the EP for two far-off-resonant mechanical modes. In fact, as shown in Eqs. (7) and (8), the optomechanically induced mechanical frequency shifts and effective mechanical coupling strength depend not only on the optical coupling strength κ but also on the optomechanical coupling strength. When we approach EP in the PT-symmetric regime, the effective optomechanical coupling strength will be greatly enhanced due to the topological-singularity-induced amplification of optomechanical nonlinearity in the vicinity of the EP [2934], although the optical coupling strength κ decreases. The increase of the effective optomechanical coupling strength compensates the decrease of the inter-cavity optical coupling strength and thus results in the enhancement of optomechanically induced mechanical frequency shifts and effective mechanical coupling strength simultaneously, which finally induces the large modification of the effective mechanical frequencies of the two resonators. This is the mechanism where weak coupling strength is not detrimental to synchronization, but benefits to it in the PT-symmetric regime.

To give more insights into the phenomena shown, we plot in Fig. 4(b) the cross-correlation function Mcc between the two mechanical displacements x1 and x2 with different inter-cavity optical coupling strength κ, where Mcc is defined as [4652]

Mcc=max0<t<+1ϕ1ϕ20+x1(τt)x2(τ)dτ,ϕi=0+xi2(τ)dτ.

This normalized cross-correlation function varies between 0 and 1. The maximum value of Mcc=1 indicates that the two time series of the mechanical displacements x1 and x2 have exactly the same shape, even though their amplitudes may be different, which implies that the two self-sustained oscillators have the same frequency, that is, the onset of synchronization. As shown in Fig. 4(b), in the PT-symmetric regime, smaller κ induces higher value of Mcc (the red solid curve), and Mcc reaches the maximum value (the unit) as κ decreases and approaches EP, which means that the two mechanical displacements x1 and x2 tend to be synchronized with the decrease of the inter-cavity coupling strength. However, in the broken-PT-symmetric regime, the cross-correlation function [the blue dashed curve in Fig. 4(b)] increases and tends to unit with the increase of κ, which means that stronger inter-cavity coupling strength will be helpful for synchronization as expected.

4. NOISE-ENHANCED SYNCHRONIZATION IN PT-SYMMETRIC OPTOMECHANICAL SYSTEM

A. Stochastic Noises in the Optical Modes

We now study the effects of the stochastic noises on our PT-symmetric optomechanical system. Two independently identically distributed Gaussian white noises ξ1,2 are introduced for the two optical modes α1,2, such that ξi(t)ξj(t+τ)=2Dδijδ(τ), where D is the intensity of the noises and small to protect the PT-symmetric structure. Here, we have included the shifts of damping rates induced by stochastic noises into the gain (γ1) and loss (γ2) rates in our optomechanical system. Thus, the dynamical equations of our PT-symmetric system can be reexpressed as

α˙k=i(Δk+gomxk)αk+(1)1+kγkαkiκα3k+2γkexϵk+ξk(t),x¨k=2Γmkx˙kΩk2xkgom|αk|2,
where k=1,2.

We present the numerical results of the cross-correlation function Mcc between the two mechanical oscillators in Figs. 5(a) and 5(c) by changing the noise strength D and fixing other parameters both in broken-PT-symmetric and PT-symmetric regimes. In order to show the benefit to the synchronization better, we choose the optical coupling strength as κ=27.76MHz in the broken-PT-symmetric regime, since the enhancement of the degree of synchronization induced by noises is large at this coupling strength when other system parameters are fixed, as shown in Figs. 5(a) and 5(b). Similarly, in Figs. 5(c) and 5(d), we choose κ=32.19MHz in the PT-symmetric regime to exhibit advantages of noises for synchronization. It can be seen that Mcc is enhanced with increasing noise intensity D both in broken-PT-symmetric and PT-symmetric regimes, reaches the maximal values at particular noise level, and then decreases at higher noise intensity. It means that synchronization process may benefit from noises [5360] in our optomechanical PT-symmetric system if the strength of the noise is not too large. To interpret what we observe, we can see that the noises will randomly shift the frequencies of the mechanical modes, especially when we approach the EP where the effects of noises are enhanced [6164]. Since the frequencies of the two mechanical modes are far-separated, these random frequency shifts may decrease the difference between the frequencies of the two mechanical modes in a certain probability with increasing noise strength D and thus increase the cross-correlation function Mcc. When we increase the noise strength D further, the noise will be strong enough to destroy the periodic oscillation of a single mechanical oscillator and the PT-symmetric structure of the optomechanical system and thus decrease the degree of synchronization between the two mechanical oscillators. This interpretation can also be confirmed by checking the variance of Mcc versus the noise strength D [Figs. 5(b) and 5(d)]. The variance of Mcc first increases with increasing noise strength D (note that Mcc increases at the same time), which means that more noises enter the system although Mcc is increased. The variance of Mcc then decreases when we increase D further, because the value of Mcc is small in this case, and the noise-induced fluctuations in Mcc are suppressed. This is the essential reason why stochastic noises at moderate levels can benefit synchronization in the PT-symmetric optomechanical system.

 figure: Fig. 5.

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=27.76MHz. (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=32.19MHz. The variance of Mcc is presented in (d).

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We also check another index of synchronization, the Kramers rate, which is an alternative index to show the correlation between the two subsystems and suitable to describe noisy synchronized systems. When the Kramers rates of two subsystems coincide with each other, the two subsystems are well correlated [53]. Now, we calculate the Kramers rates r1 and r2 of the mechanical displacements x1 and x2, respectively. The Kramers rate is originally defined as the transition rate between neighboring potential wells of a particle caused by stochastic forces, which was first proposed by Kramers in 1940 [65].

Here, we use the mean first passage time [66,67], i.e., the average time that the particle moves from one potential well to the other well, to evaluate the Kramers rates r1 and r2 of mechanical displacements x1 and x2. We obtain the histograms of x1,2 through numerical simulation first, then search for the locations with the maximum probability of x1,2, i.e., the bottoms of the potential wells of x1,2, based on the distribution of histograms, and finally calculate the mean first passage times τ1,2, i.e., the average times of jumps between the two potential wells for each mechanical displacement. The Kramers rates r1 and r2 can then be calculated by the reciprocal of the mean first passage times τ1,2, i.e., ri=1/τi (i=1,2). The simulation results for r1 and r2 are presented in Fig. 6. It can be seen that the Kramers rates r1 and r2 get closer with the increase of the noise intensity in both broken-PT-symmetric [Fig. 6(a)] and PT-symmetric [Fig. 6(b)] regimes, which indicates that the frequencies of the mechanical displacements x1 and x2 get closer when the noise intensity D increases.

 figure: Fig. 6.

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=27.76MHz 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=32.19MHz.

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B. Thermal Noises in the Mechanical Modes

In the above analysis, we do not consider the effects of the thermal noises of the mechanical modes on synchronization. However, these thermal noises acting on the mechanical modes can also benefit the synchronization between the two mechanical modes in our PT-symmetric optomechanical system. In order to simplify the discussions, we only consider the thermal noises in the mechanical modes and do not exert extra stochastic noises to the optical modes in this subsection, and we assume that these thermal noises are white noises, based on which the Langevin equations of the mechanical modes can be expressed as

x¨1=2Γmx˙1Ω˜12x1κmechx2+Γnoise1(t),x¨2=2Γmx˙2Ω˜22x2κmechx1+Γnoise2(t),
where the constant driving terms induced by the optical modes have been included into x1,2 by a coordinate transformation. The mechanical damping rate Γm consists of the original mechanical damping rate Γmo and the noise-induced damping rate δΓm, i.e., Γm=Γmo+δΓm. The mechanical thermal noises Γnoise1 and Γnoise2 are diffusion terms with δ-correlated Gaussian distribution,
Γnoisei(t)=0,Γnoisei(t)Γnoisej(t)=4ΓmkTδ(tt),
where k is the Boltzman’s constant, and T is the temperature.

To show the beneficial effect of thermal noises on synchronization, we present the numerical results of the normalized correlation function R [68] between the two mechanical oscillators in Figs. 7(a) and 7(b) by changing the temperature T and fixing other parameters in both broken-PT-symmetric and PT-symmetric regimes, where Tr is the room temperature. In the broken-PT-symmetric regime with fixed κ=27.76MHz, R (blue-dashed curve) increases with increasing temperature T and reaches 0.61 at the room temperature Tr, which is larger than 0.48 when we disregard the thermal noises. Similarly, in the PT-symmetric regime with fixed κ=32.19MHz, R (red-solid curve) increases with increasing temperature T and reaches 0.65 at room temperature, which is larger than 0.51 when we omit the thermal noises. It means that the thermal noises in the mechanical modes can also benefit the synchronization between the two mechanical modes in our optomechanical PT-symmetric system.

 figure: Fig. 7.

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=27.76MHz. (b) Effects of the thermal noises on R versus T in the PT-symmetric regime with κ=32.19MHz.

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To give more insights into the phenomenon presented, we calculate the Kramers rates r1 and r2 of mechanical displacements x1 and x2. The simulation results for Kramers rates r1 and r2 are illustrated in Fig. 8. In Fig. 8(a), the red (blue) curve denotes Kramers rate r1 (r2) with different values of temperature T in the broken-PT-symmetric regime, where optical coupling strength is κ=27.76MHz. We can see in Fig. 8(a) that Kramers rates r1 and r2 tend to get closer to each other as the temperature T increases to the room temperature Tr. A similar phenomenon can be observed in the PT-symmetric regime, as shown in Fig. 8(b), i.e., the mechanical thermal noises tend to decrease the difference between the Kramers rates r1 and r2 as the temperature increases to room temperature, where the optical coupling strength is fixed as κ=32.19MHz. These simulation results indicate that mechanical thermal noises can make the frequencies of the two mechanical displacements x1 and x2 tend to be consistent with each other and thus benefit the synchronization in our PT-symmetric optommechanical system.

 figure: Fig. 8.

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=27.76MHz is fixed. (b) The Kramers rates r1 and r2 versus the temperature T correspond to the PT-symmetric regime (κ=32.19MHz).

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Furthermore, we can also observe the beneficial effect of the mechanical thermal noises on the synchronization by theoretically analyzing the correlation function between the two mechanical modes. Actually, at small time limit [68], the normalized correlation function between the two mechanical modes can be approximately calculated by (derivation, see Appendix G)

R(τ,t)12Ω˜12τt+q2κmechΩ˜12τt2+q3κmechΩ˜12τt312Ω˜12τt+2ΓmkTκmechΩ˜12τt2+43ΓmkTκmechΩ˜12τt3,
where q=4ΓmkT is the intensity of the mechanical thermal noises. It is shown in Eq. (13) that the normalized correlation function R can be enhanced by increasing the intensity q of the thermal noises, which is consistent with the above simulation results, as shown in Figs. 7 and 8.

5. CONCLUSIONS

We have shown that the mechanical motions of two coupled PT-symmetric optomechanical resonators with far-off-resonant mechanical frequencies can be synchronized when the system approaches the EP. In particular, in the PT-symmetric regime, the two mechanical modes are easier to be synchronized with weaker optical coupling strength between the two optomechanical resonators. Additionally, it is shown that noises will be enhanced in the vicinity of the EP in our system, and the enhanced noises will benefit the synchronization process if only the strengths of the noises are not too strong. Our study opens up a new dimension of research for PT-symmetric optomechanical systems for possible applications such as metrology, cooling, and communication. It also gives new perspectives for synchronization in optomechanical systems.

APPENDIX A: PT-SYMMETRIC OPTOMECHANICAL SYSTEM WITH Δ=|Δ2Δ1|Δ1,Δ2

Generally, in our optomechanical system, if we consider symmetric optical frequency detunings Δ1=Δ2, an EP exists, where the two optical supermodes degenerate with each other at this point. However, if Δ=|Δ2Δ1|0, the degeneracy of the optical supermodes at the previous EP will be broken. Now, we demonstrate that if Δ is appropriately chosen, this non-degeneracy will be small, and the properties of the PT-symmetric structure will be maintained in our PT-symmetric optomechanical system.

In order to analyze the PT-symmetric structure in our optomechanical system, we only consider the optical modes and assume that the nonlinear optomechanical interaction between the optical mode and mechanical mode only leads to an optomechanics-induced frequency shift for the corresponding optical mode in each cavity [32]. Here, we treat the stationary state β1s (β2s) of the mechanical mode β1 (β2) as a parameter that leads to a frequency detuning Δ1s=gom(β1s+β1s*) [Δ2s=gom(β2s+β2s*)] for the optical mode α1 (α2). By taking α˙1,2=β˙1,2=0 in Eq. (1), we can obtain the stationary states of the optical and mechanical modes, which satisfy the following equations:

0=[(1)1+kγkiΔk]αksiκα3ksigomαks(βks+βks*)+2γkexϵk,0=(Γmk+iΩk)βksigom|αks|2.

By solving the above equations, the stationary states of the mechanical modes can be expressed as

β1s=gomΩ1+iΓm1Γm12+Ω12|α1s|2,β2s=gomΩ2+iΓm2Γm22+Ω22|α2s|2,
and the stationary states of the optical modes satisfy the following equations:
[(1)1+kγki(Δk+Δks)]αksiκα3ks+2γkexϵk=0,k=1,2,
where
Δ1s=2Ω1gom2Γm12+Ω12|α1s|2,Δ2s=2Ω2gom2Γm22+Ω22|α2s|2.

By substituting the stationary states β1s and β2s into Eq. (1) and eliminating the mechanical modes, we obtain the reduced motion equations of the two coupled optical modes as follows:

α˙k=[(1)1+kγki(Δk+Δks)]αkiκα3k+2γkexϵk.

Based on Eq. (A5), we can calculate the eigenfrequencies of the optical supermodes as

ωo±=iγ+(Δ¯1+Δ¯2)±i[γ++i(Δ¯2Δ¯1)]2κ2,
where γ±=(γ1±γ2)/2 and Δ¯k=(Δk+Δks)/2. Considering the balanced gain and loss (γ1=γ2=γ), the above equation can be reduced to
ωo±=Δ¯1+Δ¯2±i[γ+i(Δ¯2Δ¯1)]2κ2.

Actually, the vacuum optomechanical coupling gom in traditional microresonators [10] is very small, which means that Δ1s and Δ2s are very tiny; thus, if Δ=Δ2Δ1 is small enough, the imaginary parts in the root signs in Eq. (A7) can be omitted, and the eigenvalues can be approximately reduced to

ωo±(Δ¯1+Δ¯2)±iγ2κ2.

It means that the two eigenvalues of optical supermodes tend to degenerate with each other at the EP (κ=γ) when Δ is small enough. In fact, in order to approximately evaluate stationary states α1s and α2s, we can neglect Δ1s and Δ2s in Eq. (A3) and thus obtain α1s and α2s, since Δ1s and Δ2s are far smaller than Δ1 and Δ2. By substituting α1s and α2s into Eqs. (A4) and (A7), the non-degeneracy of the optical supermodes at the EP (κ=γ) can be approximately expressed as

Δsplitγ2|Δ+4gom2Ω2Ω1Ω1Ω21Δ2ϵ2|1γ,
under the condition
gomΔ<Δ1,2γ,
where Δsplit=Re[ωo+ωo]Im[ωo+ωo]. Equation (A9) can also be further simplified to Δsplit/γ3Δ/γ under the condition gom/Δγ,κ. It can be inferred that this non-degeneracy will be small, and the properties of the PT-symmetric structure will be maintained in our optomechanical system if the value of Δ is appropriately chosen between gom and Δ1,2.

As for the simulation results in Figs. 2(a) and 2(b) in the main text, we first calculate the stationary states of α1s and α2s by numerically solving Eq. (A3) and then obtain the eigenvalues of the optical supermodes by substituting α1s and α2s into Eqs. (A4) and (A7).

APPENDIX B: THE DERIVATION OF THE REDUCED DYNAMICAL EQUATIONS OF THE MECHANICAL MODES

Based on dynamical Eq. (1), we can adiabatically eliminate the degrees of freedom of the optical modes under the condition that the optical decay rates are far larger than the mechanical decay rates and derive the reduced dynamical equations of the mechanical modes. In fact, by rewriting the first two equations in Eq. (1) in matrix format, we have

[α˙1α˙2]=M[α1α2]+[igomα1(β1+β1*)igomα2(β2+β2*)]+[2γ1exϵ12γ2exϵ2],
where
M=[γ1iΔ1iκiκγ2iΔ2].

The matrix M can be diagonalized as

M=TΛT1,
where
Λ=[ω+00ω],T=[τ+τ11],
and
ω±=γiΔ+iκ2+(Δ+iγ+)2,τ±=Δ+iγ+κ±1+(Δ+iγ+κ)2,
where γ±=(γ1±γ2)/2 and Δ±=(Δ1±Δ2)/2. Thus, we can introduce the following optical supermodes:
[α+α]=T1[α1α2],
by which Eq. (B1) can be re-expressed as
[α˙+α˙]=[ω+00ω][α+α]igom[(λ+α++λα)(β1+β1*)λ(α++α)(β2+β2*)(λ+α++λα)(β1+β1*)+λ+(α++α)(β2+β2*)]+[μ2γ1exϵ1λ2γ2exϵ2μ2γ1exϵ1+λ+2γ2exϵ2],
where
λ±=Δ1Δ2+i(γ1+γ2)±Ξ12Ξ1,μ=κΞ1,
and
Ξ1=4κ2+[Δ1Δ2+i(γ1+γ2)]2.

To adiabatically eliminate the degrees of freedom of the optical modes, we let α˙+=α˙=0, by which we can obtain the following stationary solution:

α±ss=μ[ωigom(β2+β2*)]2γ1exϵ1Ξ2±λ[ωigom(β1+β1*)]2γ2exϵ2Ξ2,
where
Ξ2=ω+ω+igom(ω+λωλ+)(β1+β1*)+igom(ω+λ++ωλ+)(β2+β2*)gom2(λ+λ)2(β1+β1*)(β2+β2*).

By introducing the power-series expansion and omitting high-order terms of β1 and β2 since gom|Δ2Δ1|, the above solutions can be simplified as

α+ssμ2γ1exϵ1λ2γ2exϵ2ω++igomΞ3+λω+ω2γ2exϵ2(ω+ω)2(β1+β1*)+igomΞ3+μω+ω2γ1exϵ1(ω+ω)2(β2+β2*),αssμ2γ1exϵ1λ+2γ2exϵ2ω+igomΞ4++λ+ω+ω2γ2exϵ2(ω+ω)2(β1+β1*)+igomΞ4μω+ω2γ1exϵ1(ω+ω)2(β2+β2*),
where
Ξ3±=ω(μ2γ1exϵ1λ2γ2exϵ2)(ω+λωλ±),Ξ4±=ω+(μ2γ1exϵ1λ+2γ2exϵ2)(ω+λωλ±).

Thus, the stationary solutions of α1 and α2 can be expressed as

α1ss=τ+α+ss+ταss=(σ2e1iκe2)Cigom(σ22e1iκσ2e2)C2(β1+β1*)+igom(κ2e1+iκσ1e2)C2(β2+β2*),α2ss=α+ss+αss=(iκe1+σ1e2)C+igom(iκσ2e1+κ2e2)C2(β1+β1*)igom(iκσ1e1+σ12e2)C2(β2+β2*),
where
C=1/(κ2+δ2+σ2),ek=2γkexϵk,
and
σ1=γ1+iΔ1,σ2=γ2+iΔ2,δ=Δ1Δ22+iγ1+γ22,σ=γ1+γ22+iΔ1+Δ22.

By substituting the above stationary solutions α1ss and α2ss into the dynamical equations of the mechanical modes β1 and β2 in Eq. (1), and dropping the counter-rotating terms with β1,2*, the dynamical equations of the reduced mechanical system can be expressed in the matrix format as

[β˙1β˙2]=[Γm1i(Ω1+δΩ1)κmechκmechΓm2i(Ω2+δΩ2)]×[β1β2][iη1iη2],
where
δΩk=4gom2[(κ2Δ1Δ2)Δ3kΔkγ3k2]f2×[γ3k2γkexϵk2+(Δ3kγkexϵkκγ3kexϵ3k)2],κmech=4gom2κ{κγ1exϵ12[Δ2(κ2Δ1Δ22γ1γ2)+Δ1γ22]f2+κγ2exϵ22[Δ1(κ2Δ1Δ2)Δ2γ12]f2γ1exγ2exϵ1ϵ2f2[(κ2γ1γ2)2Δ12Δ22(Δ12γ22Δ22γ12)]},ηk=gom[γ3k2ϵk2+(Δ3kϵkκϵ3k)2]f,
where f=1/[(κ2Δ1Δ2γ1γ2)2+(Δ1γ2Δ2γ1)2].

APPENDIX C: THE OPTOMECHANICS-INDUCED MECHANICAL FREQUENCY SHIFTS AND EFFECTIVE MECHANICAL COUPLING

In our PT-symmetric system, we consider well-balanced gain and loss such that γ1=γ2γ and also consider the critical coupling case such that γ1ex=γ2ex=γ/2. When gom|Δ1Δ2|<Δ1,2κ,γ and ϵ1=ϵ2ϵ, the two mechanical frequency shifts δΩ1,2 in Eq. (B7) can be simplified as

δΩ1=δΩ2gom2Δ(γ2+κ2)2γϵ2[(κ2γ2)2+γ2Δ2]2.

We show the optomechanics-induced mechanical frequency shifts δΩ1,2 in Fig. 9(a). When the system is far away from the EP, the mechanical frequency shifts δΩ1 (red solid line) and δΩ2 (red dashed line) are negligibly small in comparison with the mechanical frequencies Ω1,2. However, both frequency shifts δΩ1 and δΩ2 will be greatly amplified in the vicinity of the EP, which will modify the mechanical frequencies Ω1,2 such that the frequencies of the mechanical modes b1,2 tend to overlap. As shown in Fig. 9(a), in the PT-symmetric regime (pink area) of the optical modes, these mechanical frequency shifts are enhanced with the decrease of the optical coupling strength κ, which means that smaller coupling strength κ between two optical modes is better for synchronization. In addition, as shown in Fig. 9(a), the difference between the detuning frequencies of the two optical modes, i.e., |Δ2Δ1|, significantly influences the amplification of the mechanical frequency shifts δΩ1,2 when the system is around the EP. By fixing Δ2=5MHz, we plot the curves of δΩ1,2 for different Δ1. We can see that the mechanical frequency shifts δΩ1,2 are greatly enhanced with the decrease of |Δ2Δ1| in the vicinity of the EP.

 figure: Fig. 9.

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=5MHz 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.

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Under the same assumptions, the strength of the effective mechanical coupling in Eq. (B7) can be simplified as

κmech4gom2Δκ2γ3ϵ2[(κ2γ2)2+γ2Δ2]2,
and thus the effective mechanical coupling will be greatly amplified in the vicinity of the EP. We then plot the curves of the effective mechanical coupling strength κmech versus the optical coupling strength κ in Fig. 9(b). Here, we also fix Δ2=5MHz and tune the detuning frequency Δ1. It can be seen that the effective mechanical coupling strength κmech is significantly enhanced in the vicinity of the EP. Therefore, in the PT-symmetric regime, weaker optical coupling strength leads to stronger effective mechanical coupling strength and thus may be helpful for the synchronization between the two mechanical modes. It is also shown that the degree of amplification of κmech is extensively enhanced with the decreasing of |Δ2Δ1| in the vicinity of the EP.

APPENDIX D: THE INFLUENCE OF THE EFFECTIVE MECHANICAL COUPLING STRENGTH ON SYNCHRONIZATION

In this part, we discuss the positive effect of the enhancement of the effective mechanical coupling κmech on the synchronization between mechanical modes, i.e., the stronger the κmech is, the easier the synchronization is. For simplicity and clarity, we re-express the dynamical equation in Eq. (B6) by using the differential operator format as follows:

{D+[Γm1+i(Ω1+δΩ1)]}β1+κmechβ2=iη1,κmechβ1+{D+[Γm2+i(Ω2+δΩ2)]}β2=iη2,
where D represents the differential operator. By eliminating the degree of freedom of β2, we can derive the dynamical equation of β1 and then obtain the characteristic equation of this coupled system as follows:
λ2+[Γm1+Γm2+i(Ω1+δΩ1+Ω2+δΩ2)]λ+[Γm1+i(Ω1+δΩ1)][Γm2+i(Ω2+δΩ2)]κmech2=0.

By considering Γm1=Γm2=Γm, the roots of this characteristic equation can be expressed as

λ±=ΓmiΩAve+±iΩAve2κmech2,
where
ΩAve±=Ω1+δΩ1±(Ω2+δΩ2)2.

It can be easily seen that in the weak coupling regime such that κmech<ΩAve, the vibration frequencies of the mechanical modes β1,2 get closer to each other with the increase of the effective coupling strength κmech, which means that the degree of synchronization between the two mechanical modes increases with increasing κmech. At the critical point such that κmech=ΩAve, the two oscillators will have the same vibration frequency ΩAve+, which means that the frequency synchronization between the two mechanical modes is accomplished, i.e., the frequencies of the two mechanical modes are equal to each other. It is shown that a stronger effective mechanical coupling strength can improve the degree of synchronization between the mechanical modes in our system and thus leads to frequency synchronization when the effective mechanical coupling is strong enough.

In addition, in the weak coupling regime, Eq. (D1) can also be re-expressed as

λ+=Γmi(2Ω˜2δΩcoup),λ=Γmi(2Ω˜1+δΩcoup),
where Ω˜k=(Ωk+δΩk)/2 and
δΩcoup=Ω˜2Ω˜1(Ω˜2Ω˜1)2κmech2
is induced by the effective mechanical coupling strength κmech. In other words, the effective frequencies of the two mechanical modes can be given by
Ω1,eff=Ω1+δΩ1+δΩcoup,Ω2,eff=Ω2+δΩ2δΩcoup,
where Ω1,eff (Ω2,eff) is the effective frequency of mechanical mode β1 (β2). It can be seen in Eq. (D4) that both optomechanics-induced mechanical frequency shift δΩi and effective mechanical coupling κmech can result in frequency shifts for the two mechanical modes and thus contribute to the synchronization together.

APPENDIX E: THE ENHANCEMENT OF THE OPTOMECHANICAL INTERACTION IN PT-SYMMETRIC OPTOMECHANICAL SYSTEM

In our PT-symmetric optomechanical system, an enhancement of the effective optomechanical coupling due to the topological-singularity-induced amplification of optomechanical nonlinearity in the vicinity of the EP exists [29,64]. This enhanced optomechanical coupling then leads to the amplification of the optomechanics-induced mechanical frequency shifts δΩ1,2 and the effective mechanical coupling strength κmech. Since both the optomechanics-induced mechanical frequency shifts and the effective mechanical coupling can change the frequencies of the two mechanical modes, the synchronization between far-off-resonant mechanical modes may be realized with sufficiently large optomechanical coupling strength. In the PT-symmetric regime, the system gets close to the EP with the decrease of the optical coupling strength κ, which results in an enhancement of the optomechanical coupling and thus compensates the reduction of the inter-cavity optical coupling strength. In the following part of this subsection, we will show that the optomechanical coupling can be largely enhanced in our PT-symmetric system.

In our optomechanical system, the interaction Hamiltonian between optical modes and mechanical modes can be expressed as

Hint=goma1a1(b1+b1)+goma2a2(b2+b2),
where a1 (a2) and b1 (b2) represent the annihilation operators of the optical mode and mechanical mode in the active (passive) resonator, respectively, and gom is the vacuum optomechanical coupling strength. If we re-write this interaction Hamiltonian Hint in the optical supermodes picture, then the effective optomechanical coupling strength geff between optical supermodes and mechanical modes can be expressed as
geffgom2[γ2(κ2γ2)2+γ2Δ2+1].

Since Δ=|Δ2Δ1|κ,γ, the effective optomechanical coupling strength geff can be greatly amplified in the vicinity of the EP when κγ. We show the simulation results of the effective optomechanical coupling strength geff versus the optical coupling strength κ in Fig. 10. When the optical coupling strength κ is far away from the EP, i.e., in the green areas in Fig. 10, the effective optomechanical coupling strength changes slowly with the optical coupling strength κ. However, in the pink area, geff increases very fast when the system approaches the EP. In addition, by comparing Eq. (E2) with Eqs. (C1) and (C2), we can find that |δΩ1,2|geff4|f1(κ,γ,ϵ,gom,Δ)| and κmechgeff4|f2(κ,γ,ϵ,gom,Δ)|, which means that the enhanced optomechanical coupling strength can lead to improvements of the optomechanics-induced mechanical frequency shifts and the effective mechanical coupling in the vicinity of the EP. It should be pointed out that the optical spring effect, i.e., the mechanical frequency shift induced by the optical field, can be increased in the strong optomechanical coupling regime, which is beneficial to the synchronization between two mechanical resonators. However, in our scheme, as the effective optomechanical coupling strength can be greatly enhanced due to the topological-singularity-induced amplification of the PT-symmetric structure, we do not require the strict condition of strong coupling regime in our scheme, which minimizes the restrictions on synchronization in experiment.

 figure: Fig. 10.

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.

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APPENDIX F: THE DIFFERENCE BETWEEN ACTIVE PT-SYMMETRIC SYSTEM AND PASSIVE SYSTEM WITH EP FOR SYNCHRONIZATION

Based on the previous discussion, we know that in the discussed gain–loss balanced PT-symmetric optomechanical system there exists amplifications of the optomechanics-induced mechanical frequency shifts and effective mechanical coupling strength in the vicinity of the EP. However, if this PT-symmetric system is replaced by a passive coupled system with an EP, i.e., the active resonator in the discussed PT-symmetric system is replaced by a passive resonator, the two far-detuned mechanical modes in this system will not be synchronized. To show this, we can easily obtain the dynamical equations of the system by replacing the optical damping γ1 in Eq. (1) with γ1 as follows:

α˙k=(γkiΔk)αkiκα3kigomαk(βk+βk*)+2γkexϵk,β˙k=(Γmk+iΩk)βkigom|αk|2.

Under the assumptions that 2γ1exϵ1=2γ2exϵ2=ϵ and |Δ2Δ1|<Δ1,2κ,γ1,2, the optomechanics-induced mechanical frequency shifts δΩ1,2 and the effective mechanical coupling κmech can be approximately expressed as

δΩ1=δΩ2gom2Δϵ2[(κ2+γ1γ2)+Δ+2]2,κmech2δΩ1κ/Δ,
where Δ+=(Δ1+Δ2)/2 and Δ=Δ2Δ1. As Δκ,γ1,2, and gom is very tiny, δΩ1,2 and κmech are very small. This implies that in this passive system with an EP, the amplifications of mechanical frequency shifts and effective mechanical coupling are not strong enough. Thus, these two mechanical modes with far-off-resonant mechanical frequencies cannot be synchronized.

In addition, if the balance between gain and loss is broken in our PT-symmetric system, i.e., Γ=|γ1γ2|/20, the synchronization between the two mechanical modes will be suppressed. In fact, when the balance between gain and loss is broken, the mechanical frequency shifts δΩ1,2 and the effective mechanical coupling κmech can be expressed as

δΩk2gom2Δ(κ2+γ3k2)2ϵ2[(κ2γ1γ2)2+(γ1+γ2)2Δ2/4+Γ2Δ+2]2,κmech2δΩ2κ2γ1γ2κ2+γ12.

Therefore, with the increase of Γ, the amplification effects of the mechanical frequency shifts and the effective mechanical coupling strength will be suppressed. We show the mechanical frequency shifts δΩ1,2 and the effective mechanical coupling strength κmech with different Γ in Figs. 11(a), 11(b), and 11(c), respectively. It can be clearly seen that the amplifications of the mechanical frequency shifts and the effective mechanical coupling strength are seriously suppressed when Γ is large; thus, the synchronization between the two mechanical modes with far-off resonance is difficult.

 figure: Fig. 11.

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.

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APPENDIX G: NORMALIZED CORRELATION FUNCTION IN PT-SYMMETRIC OPTOMECHANICAL SYSTEM WITH THERMAL NOISES

In this subsection, we will discuss the influence of the thermal noises in mechanical modes on the normalized correlation function R between the two mechanical modes. To simplify our discussions, we redefine four variables,

ξ1=x1,ξ2=ξ˙1,ξ3=x2,ξ4=ξ˙3;
thus, the Langevin equation of the mechanical modes in Eq. (7) in the main text can be re-expressed as
[ξ˙1(t)ξ˙2(t)ξ˙3(t)ξ˙4(t)]=[0100Ω˜122Γmκmech00001κmech0Ω˜222Γm][ξ1(t)ξ2(t)ξ3(t)ξ4(t)]+[Γ1Γ2Γ3Γ4]=A[ξ1(t)ξ2(t)ξ3(t)ξ4(t)]+[Γ1Γ2Γ3Γ4],
where Γ1=Γ3=0, Γ2=Γnoise1, and Γ4=Γnoise2. The solution of the above matrix equation can be expressed as
ξi(t)=k=14Gik(t)zk+k=140tGik(t)Γk(tt)dt,
where matrix G=(Gij)=exp(At), and zi represents the initial values of the variables ξi. As we consider small time t, the matrix G can be approximately expressed as
G(t)=eAtIAt[1t00Ω˜12t12Γmtκmecht0001tκmecht0Ω˜22t12Γmt];
thus, the solution of ξ1(t) in Eq. (G3) can be approximately expressed as
ξ1(t)=G11z1+G12z2+G13z3+G14z4+0tG11(t)Γ1(tt)+G12(t)Γ2(tt)+G13(t)Γ3(tt)+G14(t)Γ4(tt)dt=z1+tz2+0ttΓ2(tt)dt.

Similarly, other solutions in Eq. (G3) can be approximately expressed as

ξ2(t)=(κmechz3+Ω˜12z1)t+(12Γmt)z2+0t(12Γmt)Γ2(tt)dt,ξ3(t)=z3+z4t+0ttΓ4(tt)dt,ξ4(t)=(κmechz1+Ω˜22z3)t+(12Γmt)z4+0t(12Γmt)Γ4(tt)dt.

We then calculate the correlation functions as

Rij(τ,t)=ξi(t+τ)ξj(t),
where · is the ensemble average over the stochastic noises. Based on the regression theorem [68], we know that the correlation functions Rij(τ,t) can be reduced to
Rij(τ,t)=k=14Gik(τ)ξk(t)ξj(t),0τ.

By substituting the solutions of ξi(t), as shown in Eqs. (G7) and (G8), into the correlation functions Rij(τ,t) [Eq. (G9)], the three correlation functions R13(τ,t), R11(0,t), and R33(0,t) can be expressed as

R13(τ,t)=(z1+tz2)(z3+z4t)+q3t3+τ[(κmechz3+Ω˜12z1)(z3+z4t)t+(12Γmt)(z3+z4t)z2+q(12t223Γmt3)],R11(0,t)=(z1+z2t)2+q3t3,R33(0,t)=(z3+z4t)2+q3t3.

For simplicity, we assume that the system is stationary at the initial time, i.e., z2=z4=0 and consider the case where z1=1/Ω˜12,z3=1/κmech; thus, the normalized correlation function between the two mechanical modes can be approximately expressed as

R(τ,t)=|R13(τ,t)|R11(0,t)R33(0,t)=|12Ω˜12τt+qκmechΩ˜122τt2+qκmechΩ˜123τt3|1+q3κmech2t31+q3Ω˜14t312Ω˜12τt+q2κmechΩ˜12τt2+q3κmechΩ˜12τt312Ω˜12τt+2ΓmkTκmechΩ˜12τt2+43ΓmkTκmechΩ˜12τt3.

Funding

National Natural Science Foundation of China (11674194, 12004044); National Defense Basic Scientific Research Program of China (JCKY2019407C002); National Key Research and Development Program of China (2014CB921401); Tsinghua Initiative Scientific Research Program; Tsinghua National Laboratory for Information Science and Technology; Beijing National Research Center for Information Science and Technology; National Science Foundation (EFMA1641109); Army Research Office (W911NF1210026, W911NF1710189); Joint Fund of Science & Technology Department of Liaoning Province and State Key Laboratory of Robotics, China (2021-KF-22-01); Guoqiang Research Institute of Tsinghua University (20212000704).

Disclosures

The authors declare no conflicts of interest.

REFERENCES

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4. A. T. Winfree, The Geometry of Biological Time, 2nd ed. (Springer, 2001).

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6. A. F. Taylor, M. R. Tinsley, F. Wang, Z. Y. Huang, and K. Showalter, “Dynamical quorum sensing and synchronization in large populations of chemical oscillators,” Science 323, 614–617 (2009). [CrossRef]  

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8. S. C. Manrubia, A. S. Mikhailov, and D. H. Zanette, Emergence of Dynamical Order: Synchronization Phenomena in Complex Systems (World Scientific, 2004).

9. S. Bregni, Synchronization of Digital Telecommunications Networks (Wiley, 2002).

10. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014). [CrossRef]  

11. C. A. Holmes, C. P. Meaney, and G. J. Milburn, “Synchronization of many nanomechanical resonators coupled via a common cavity field,” Phys. Rev. E. 85, 066203 (2012). [CrossRef]  

12. T. Li, T. Y. Bao, Y. L. Zhang, C. L. Zou, X. B. Zou, and G. C. Guo, “Long-distance synchronization of unidirectionally cascaded optomechanical systems,” Opt. Express 24, 12338–12348 (2016). [CrossRef]  

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27. G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in PT-symmetric optical systems,” Phys. Rev. A 85, 031802 (2012). [CrossRef]  

28. J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014). [CrossRef]  

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30. H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X. Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015). [CrossRef]  

31. Z. P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X. Y. Lü, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016). [CrossRef]  

32. D. W. Schönleber, A. Eisfelf, and R. El-Ganainy, “Optomechanical interactions in non-Hermitian photonic molecules,” New J. Phys. 18, 045014 (2016). [CrossRef]  

33. X. Y. Lü, H. Jing, J. Y. Ma, and Y. Wu, “PT-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015). [CrossRef]  

34. J. Zhang, P. Bo, S. K. Özdemir, Y. X. H. Jing, X. Y. Lü, Y. L. Liu, L. Yang, and F. Nori, “Giant nonlinearity via breaking parity-time symmetry: a route to low-threshold phonon diodes,” Phys. Rev. B 92, 115407 (2015). [CrossRef]  

35. B. Peng, S. K. Özdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014). [CrossRef]  

36. B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss induced suppression and revival of lasing,” Science 346, 328–332 (2014). [CrossRef]  

37. L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Singlemode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014). [CrossRef]  

38. H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014). [CrossRef]  

39. L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014). [CrossRef]  

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41. J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016). [CrossRef]  

42. W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017). [CrossRef]  

43. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017). [CrossRef]  

44. X. Zhou and Y. D. Chong, “PT symmetry breaking and nonlinear optical isolation in coupled imcrocavities,” Opt. Express 24, 6916–6930 (2016). [CrossRef]  

45. A. U. Hassan, H. Hodaei, M. A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92, 063807 (2015). [CrossRef]  

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49. P. H. White, “Cross correlation in structural systems: dispersion and nondispersion waves,” J. Acoust. Soc. Am. 45, 1118–1128 (1969). [CrossRef]  

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51. J. P. Lewis, “Fast normalized cross-correlation,” in Proc. Vision Interface (1995), p. 120.

52. J. C. Yoo and T. H. Han, “Fast normalized cross-correlation,” Circuits Syst. Signal Process. 28, 819 (2009). [CrossRef]  

53. A. Neiman, “Synchronization like phenomena in coupled stochastic bistable systems,” Phys. Rev. E 49, 3484–3487 (1994). [CrossRef]  

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References

  • View by:

  1. A. S. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Unified Approach to Nolinear Science (Cambridge University, 2001).
  2. R. Brown and L. Kocarev, “A unifying definition of synchronization for dynamical systems,” Chaos 10, 344–349 (2000).
    [Crossref]
  3. L. Glass and M. C. Mackey, From Clocks to Chaos: The Rhythms of Life (Princeton University, 1988).
  4. A. T. Winfree, The Geometry of Biological Time, 2nd ed. (Springer, 2001).
  5. A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour (Cambridge University, 1996).
  6. A. F. Taylor, M. R. Tinsley, F. Wang, Z. Y. Huang, and K. Showalter, “Dynamical quorum sensing and synchronization in large populations of chemical oscillators,” Science 323, 614–617 (2009).
    [Crossref]
  7. S. H. Strogatz, Sync: The Emerging Science of Spontaneous Order (Hyperion, 2003).
  8. S. C. Manrubia, A. S. Mikhailov, and D. H. Zanette, Emergence of Dynamical Order: Synchronization Phenomena in Complex Systems (World Scientific, 2004).
  9. S. Bregni, Synchronization of Digital Telecommunications Networks (Wiley, 2002).
  10. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
    [Crossref]
  11. C. A. Holmes, C. P. Meaney, and G. J. Milburn, “Synchronization of many nanomechanical resonators coupled via a common cavity field,” Phys. Rev. E. 85, 066203 (2012).
    [Crossref]
  12. T. Li, T. Y. Bao, Y. L. Zhang, C. L. Zou, X. B. Zou, and G. C. Guo, “Long-distance synchronization of unidirectionally cascaded optomechanical systems,” Opt. Express 24, 12338–12348 (2016).
    [Crossref]
  13. M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109, 233906 (2012).
    [Crossref]
  14. M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic cavity synchronization of nanomechanical oscillators,” Phys. Rev. Lett. 111, 213902 (2013).
    [Crossref]
  15. S. Y. Shah, M. Zhang, R. Rand, and M. Lipson, “Master-slave locking of optomechanical oscillators over a long distance,” Phys. Rev. Lett. 114, 113602 (2015).
    [Crossref]
  16. N. Yang, A. Miranowicz, Y. C. Liu, K. Xia, and F. Nori, “Chaotic synchronization of two optical modes in optomechanical systems,” Sci. Rep. 9, 15874 (2019).
    [Crossref]
  17. E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482, 63–67 (2012).
    [Crossref]
  18. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
    [Crossref]
  19. A. Mostafazadeh, “Pseudo-hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 205–214 (2002).
    [Crossref]
  20. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
    [Crossref]
  21. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
    [Crossref]
  22. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [Crossref]
  23. C. T. West, T. Kottos, and T. Prosen, “PT-symmetric wave chaos,” Phys. Rev. Lett. 104, 054102 (2010).
    [Crossref]
  24. L. Feng, M. Ayache, J. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
    [Crossref]
  25. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
    [Crossref]
  26. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
    [Crossref]
  27. G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in PT-symmetric optical systems,” Phys. Rev. A 85, 031802 (2012).
    [Crossref]
  28. J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
    [Crossref]
  29. H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
    [Crossref]
  30. H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X. Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
    [Crossref]
  31. Z. P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X. Y. Lü, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
    [Crossref]
  32. D. W. Schönleber, A. Eisfelf, and R. El-Ganainy, “Optomechanical interactions in non-Hermitian photonic molecules,” New J. Phys. 18, 045014 (2016).
    [Crossref]
  33. X. Y. Lü, H. Jing, J. Y. Ma, and Y. Wu, “PT-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
    [Crossref]
  34. J. Zhang, P. Bo, S. K. Özdemir, Y. X. H. Jing, X. Y. Lü, Y. L. Liu, L. Yang, and F. Nori, “Giant nonlinearity via breaking parity-time symmetry: a route to low-threshold phonon diodes,” Phys. Rev. B 92, 115407 (2015).
    [Crossref]
  35. B. Peng, S. K. Özdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
    [Crossref]
  36. B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss induced suppression and revival of lasing,” Science 346, 328–332 (2014).
    [Crossref]
  37. L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Singlemode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
    [Crossref]
  38. H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
    [Crossref]
  39. L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
    [Crossref]
  40. H. Xu, D. Mason, L. Jiang, and G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
    [Crossref]
  41. J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
    [Crossref]
  42. W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
    [Crossref]
  43. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
    [Crossref]
  44. X. Zhou and Y. D. Chong, “PT symmetry breaking and nonlinear optical isolation in coupled imcrocavities,” Opt. Express 24, 6916–6930 (2016).
    [Crossref]
  45. A. U. Hassan, H. Hodaei, M. A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92, 063807 (2015).
    [Crossref]
  46. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1978).
  47. L. R. Rabiner and R. W. Schaefer, Digital Processing of Speech Signals (Prentice-Hall, 1978).
  48. N. A. Anstey, “Correlation techniques–a review,” Can. J. Expl. Geophys. 2, 55–82 (1966).
  49. P. H. White, “Cross correlation in structural systems: dispersion and nondispersion waves,” J. Acoust. Soc. Am. 45, 1118–1128 (1969).
    [Crossref]
  50. M. V. Heel, “Similarity measures between images,” Ultrmicroscopy 21, 95–100 (1987).
    [Crossref]
  51. J. P. Lewis, “Fast normalized cross-correlation,” in Proc. Vision Interface (1995), p. 120.
  52. J. C. Yoo and T. H. Han, “Fast normalized cross-correlation,” Circuits Syst. Signal Process. 28, 819 (2009).
    [Crossref]
  53. A. Neiman, “Synchronization like phenomena in coupled stochastic bistable systems,” Phys. Rev. E 49, 3484–3487 (1994).
    [Crossref]
  54. S. K. Han, T. G. Yim, D. E. Postnov, and O. V. Sosnovtseva, “Interacting coherence resonance oscillators,” Phys. Rev. Lett. 83, 1771–1774 (1999).
    [Crossref]
  55. A. Neiman, L. Schimansky-Geier, A. Cornell-Bell, and F. Moss, “Noise-enhanced phase synchronization in excitable media,” Phys. Rev. Lett. 83, 4896–4899 (1999).
    [Crossref]
  56. H. Nakao, K. Arai, and Y. Kawamura, “Noise-induced synchronization and clustering in ensembles of uncoupled limit-cycle oscillators,” Phys. Rev. Lett. 98, 184101 (2007).
    [Crossref]
  57. K. H. Nagai and H. Kori, “Noise-induced synchronization of a large population of globally coupled nonidentical oscillators,” Phys. Rev. E 81, 065202 (2010).
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2019 (1)

N. Yang, A. Miranowicz, Y. C. Liu, K. Xia, and F. Nori, “Chaotic synchronization of two optical modes in optomechanical systems,” Sci. Rep. 9, 15874 (2019).
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2018 (1)

J. Zhang, B. Peng, S. K. Özdemir, S. Rotter, K. Pichler, D. Krimer, G. Zhao, F. Nori, Y.-X. Liu, and L. Yang, “A phonon laser operating at the exceptional point,” Nat. Photonics 12, 479–484 (2018).
[Crossref]

2017 (2)

W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
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2016 (6)

X. Zhou and Y. D. Chong, “PT symmetry breaking and nonlinear optical isolation in coupled imcrocavities,” Opt. Express 24, 6916–6930 (2016).
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T. Li, T. Y. Bao, Y. L. Zhang, C. L. Zou, X. B. Zou, and G. C. Guo, “Long-distance synchronization of unidirectionally cascaded optomechanical systems,” Opt. Express 24, 12338–12348 (2016).
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Z. P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X. Y. Lü, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref]

D. W. Schönleber, A. Eisfelf, and R. El-Ganainy, “Optomechanical interactions in non-Hermitian photonic molecules,” New J. Phys. 18, 045014 (2016).
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H. Xu, D. Mason, L. Jiang, and G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
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J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

2015 (5)

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X. Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
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X. Y. Lü, H. Jing, J. Y. Ma, and Y. Wu, “PT-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

J. Zhang, P. Bo, S. K. Özdemir, Y. X. H. Jing, X. Y. Lü, Y. L. Liu, L. Yang, and F. Nori, “Giant nonlinearity via breaking parity-time symmetry: a route to low-threshold phonon diodes,” Phys. Rev. B 92, 115407 (2015).
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S. Y. Shah, M. Zhang, R. Rand, and M. Lipson, “Master-slave locking of optomechanical oscillators over a long distance,” Phys. Rev. Lett. 114, 113602 (2015).
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A. U. Hassan, H. Hodaei, M. A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92, 063807 (2015).
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2014 (8)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
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B. Peng, S. K. Özdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
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B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Singlemode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
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H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
[Crossref]

H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref]

2013 (2)

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic cavity synchronization of nanomechanical oscillators,” Phys. Rev. Lett. 111, 213902 (2013).
[Crossref]

Y. M. Lai and M. A. Porter, “Noise-induced synchronization, desynchronization, and clustering in globally coupled nonidentical oscillators,” Phys. Rev. E 88, 012905 (2013).
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2012 (5)

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109, 233906 (2012).
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C. A. Holmes, C. P. Meaney, and G. J. Milburn, “Synchronization of many nanomechanical resonators coupled via a common cavity field,” Phys. Rev. E. 85, 066203 (2012).
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E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482, 63–67 (2012).
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A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in PT-symmetric optical systems,” Phys. Rev. A 85, 031802 (2012).
[Crossref]

2011 (3)

L. Feng, M. Ayache, J. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

G. Yoo, H.-S. Sim, and H. Schomerus, “Quantum noise and mode nonorthogonality in non-Hermitian PT-symmetric optical resonators,” Phys. Rev. A 84, 063833 (2011).
[Crossref]

2010 (4)

H. Schomerus, “Quantum noise and self-sustained radiation of PT-symmetric systems,” Phys. Rev. Lett. 104, 233601 (2010).
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K. H. Nagai and H. Kori, “Noise-induced synchronization of a large population of globally coupled nonidentical oscillators,” Phys. Rev. E 81, 065202 (2010).
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C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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C. T. West, T. Kottos, and T. Prosen, “PT-symmetric wave chaos,” Phys. Rev. Lett. 104, 054102 (2010).
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2009 (3)

A. F. Taylor, M. R. Tinsley, F. Wang, Z. Y. Huang, and K. Showalter, “Dynamical quorum sensing and synchronization in large populations of chemical oscillators,” Science 323, 614–617 (2009).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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J. C. Yoo and T. H. Han, “Fast normalized cross-correlation,” Circuits Syst. Signal Process. 28, 819 (2009).
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2008 (1)

S.-Y. Lee, J.-W. Ryu, J.-B. Shim, S.-B. Lee, S. W. Kim, and K. An, “Divergent Petermann factor of interacting resonances in a stadium-shaped microcavity,” Phys. Rev. A 78, 015805 (2008).
[Crossref]

2007 (2)

H. Nakao, K. Arai, and Y. Kawamura, “Noise-induced synchronization and clustering in ensembles of uncoupled limit-cycle oscillators,” Phys. Rev. Lett. 98, 184101 (2007).
[Crossref]

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
[Crossref]

2003 (2)

D. H. He, P. L. Shi, and L. Stone, “Noise-induced synchronization in realistic models,” Phys. Rev. E 67, 027201 (2003).
[Crossref]

H. Hofmann and F. A. Ivanyuk, “Mean first passage time for nuclear fission and the emission of light particles,” Phys. Rev. Lett. 90, 132701 (2003).
[Crossref]

2002 (2)

C. S. Zhou and J. Kurths, “Noise-induced phase synchronization and synchronization transitions in chaotic oscillators,” Phys. Rev. Lett. 88, 230602 (2002).
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A. Mostafazadeh, “Pseudo-hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 205–214 (2002).
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2000 (1)

R. Brown and L. Kocarev, “A unifying definition of synchronization for dynamical systems,” Chaos 10, 344–349 (2000).
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1999 (2)

S. K. Han, T. G. Yim, D. E. Postnov, and O. V. Sosnovtseva, “Interacting coherence resonance oscillators,” Phys. Rev. Lett. 83, 1771–1774 (1999).
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A. Neiman, L. Schimansky-Geier, A. Cornell-Bell, and F. Moss, “Noise-enhanced phase synchronization in excitable media,” Phys. Rev. Lett. 83, 4896–4899 (1999).
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1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
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1994 (1)

A. Neiman, “Synchronization like phenomena in coupled stochastic bistable systems,” Phys. Rev. E 49, 3484–3487 (1994).
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1987 (1)

M. V. Heel, “Similarity measures between images,” Ultrmicroscopy 21, 95–100 (1987).
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1969 (1)

P. H. White, “Cross correlation in structural systems: dispersion and nondispersion waves,” J. Acoust. Soc. Am. 45, 1118–1128 (1969).
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1966 (1)

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1952 (1)

G. Klein, “Mean first-passage times of Brownian motion and related problems,” Proc. R. Soc. London A 211, 431–443 (1952).
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1940 (1)

H. A. Kramers, “Brownian motion in a field of force and the diffusion model of chemical reactions,” Physica 7, 284–304 (1940).
[Crossref]

Agarwal, G. S.

G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in PT-symmetric optical systems,” Phys. Rev. A 85, 031802 (2012).
[Crossref]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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An, K.

S.-Y. Lee, J.-W. Ryu, J.-B. Shim, S.-B. Lee, S. W. Kim, and K. An, “Divergent Petermann factor of interacting resonances in a stadium-shaped microcavity,” Phys. Rev. A 78, 015805 (2008).
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Anstey, N. A.

N. A. Anstey, “Correlation techniques–a review,” Can. J. Expl. Geophys. 2, 55–82 (1966).

Arai, K.

H. Nakao, K. Arai, and Y. Kawamura, “Noise-induced synchronization and clustering in ensembles of uncoupled limit-cycle oscillators,” Phys. Rev. Lett. 98, 184101 (2007).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Ayache, M.

L. Feng, M. Ayache, J. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Bagheri, M.

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic cavity synchronization of nanomechanical oscillators,” Phys. Rev. Lett. 111, 213902 (2013).
[Crossref]

Bao, T. Y.

T. Li, T. Y. Bao, Y. L. Zhang, C. L. Zou, X. B. Zou, and G. C. Guo, “Long-distance synchronization of unidirectionally cascaded optomechanical systems,” Opt. Express 24, 12338–12348 (2016).
[Crossref]

Barnard, A.

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109, 233906 (2012).
[Crossref]

Bender, C. M.

B. Peng, S. K. Özdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Bersch, C.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

Bo, P.

J. Zhang, P. Bo, S. K. Özdemir, Y. X. H. Jing, X. Y. Lü, Y. L. Liu, L. Yang, and F. Nori, “Giant nonlinearity via breaking parity-time symmetry: a route to low-threshold phonon diodes,” Phys. Rev. B 92, 115407 (2015).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Bohm, J.

J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
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R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1978).

Bregni, S.

S. Bregni, Synchronization of Digital Telecommunications Networks (Wiley, 2002).

Brown, R.

R. Brown and L. Kocarev, “A unifying definition of synchronization for dynamical systems,” Chaos 10, 344–349 (2000).
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Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Chang, L.

L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Chen, W.

W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

Chen, Y. F.

L. Feng, M. Ayache, J. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Chong, Y. D.

Christodoulides, D. N.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

A. U. Hassan, H. Hodaei, M. A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92, 063807 (2015).
[Crossref]

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Cornell-Bell, A.

A. Neiman, L. Schimansky-Geier, A. Cornell-Bell, and F. Moss, “Noise-enhanced phase synchronization in excitable media,” Phys. Rev. Lett. 83, 4896–4899 (1999).
[Crossref]

Deléglise, S.

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482, 63–67 (2012).
[Crossref]

Doppler, J.

J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
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Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
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Eisfelf, A.

D. W. Schönleber, A. Eisfelf, and R. El-Ganainy, “Optomechanical interactions in non-Hermitian photonic molecules,” New J. Phys. 18, 045014 (2016).
[Crossref]

El-Ganainy, R.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

D. W. Schönleber, A. Eisfelf, and R. El-Ganainy, “Optomechanical interactions in non-Hermitian photonic molecules,” New J. Phys. 18, 045014 (2016).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Fainman, Y.

L. Feng, M. Ayache, J. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Fan, L.

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic cavity synchronization of nanomechanical oscillators,” Phys. Rev. Lett. 111, 213902 (2013).
[Crossref]

Fan, S. H.

B. Peng, S. K. Özdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Feng, L.

L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Singlemode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Garcia-Gracia, H.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

Geng, Z.

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X. Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
[Crossref]

Gianfreda, M.

B. Peng, S. K. Özdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

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Figures (11)

Fig. 1.
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.
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.
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.
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.
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=27.76MHz. (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=32.19MHz. The variance of Mcc is presented in (d).
Fig. 6.
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=27.76MHz 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=32.19MHz.
Fig. 7.
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=27.76MHz. (b) Effects of the thermal noises on R versus T in the PT-symmetric regime with κ=32.19MHz.
Fig. 8.
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=27.76MHz is fixed. (b) The Kramers rates r1 and r2 versus the temperature T correspond to the PT-symmetric regime (κ=32.19MHz).
Fig. 9.
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=5MHz 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.
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.
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.

Equations (65)

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α˙k=Γopkαkiκα3kigomαk(βk+βk*)+2γkexϵk,β˙k=(Γmk+iΩk)βkigom|αk|2,
ddt[α1α2]=i[2Δ¯1+iγκκ2Δ¯2iγ][α1α2]+[2γ1exϵ12γ2exϵ2],
ωo+=Δ¯1+Δ¯2±i[γ+i(Δ¯2Δ¯1)]2κ2.
ωo±=Δ1+Δ22±i(γ+iΔ2)2κ2.
gomΔ<Δ1,2γ,κ,
Δsplitγ2|Δ+4gom2Ω2Ω1Ω1Ω21Δ2ϵ2|1γ,
δΩ1=δΩ2gom2Δ(γ2+κ2)2γϵ2[(κ2γ2)2+γ2Δ2]2.
κmech4gom2Δκ2γ3ϵ2[(κ2γ2)2+γ2Δ2]2.
Mcc=max0<t<+1ϕ1ϕ20+x1(τt)x2(τ)dτ,ϕi=0+xi2(τ)dτ.
α˙k=i(Δk+gomxk)αk+(1)1+kγkαkiκα3k+2γkexϵk+ξk(t),x¨k=2Γmkx˙kΩk2xkgom|αk|2,
x¨1=2Γmx˙1Ω˜12x1κmechx2+Γnoise1(t),x¨2=2Γmx˙2Ω˜22x2κmechx1+Γnoise2(t),
Γnoisei(t)=0,Γnoisei(t)Γnoisej(t)=4ΓmkTδ(tt),
R(τ,t)12Ω˜12τt+q2κmechΩ˜12τt2+q3κmechΩ˜12τt312Ω˜12τt+2ΓmkTκmechΩ˜12τt2+43ΓmkTκmechΩ˜12τt3,
0=[(1)1+kγkiΔk]αksiκα3ksigomαks(βks+βks*)+2γkexϵk,0=(Γmk+iΩk)βksigom|αks|2.
β1s=gomΩ1+iΓm1Γm12+Ω12|α1s|2,β2s=gomΩ2+iΓm2Γm22+Ω22|α2s|2,
[(1)1+kγki(Δk+Δks)]αksiκα3ks+2γkexϵk=0,k=1,2,
Δ1s=2Ω1gom2Γm12+Ω12|α1s|2,Δ2s=2Ω2gom2Γm22+Ω22|α2s|2.
α˙k=[(1)1+kγki(Δk+Δks)]αkiκα3k+2γkexϵk.
ωo±=iγ+(Δ¯1+Δ¯2)±i[γ++i(Δ¯2Δ¯1)]2κ2,
ωo±=Δ¯1+Δ¯2±i[γ+i(Δ¯2Δ¯1)]2κ2.
ωo±(Δ¯1+Δ¯2)±iγ2κ2.
Δsplitγ2|Δ+4gom2Ω2Ω1Ω1Ω21Δ2ϵ2|1γ,
gomΔ<Δ1,2γ,
[α˙1α˙2]=M[α1α2]+[igomα1(β1+β1*)igomα2(β2+β2*)]+[2γ1exϵ12γ2exϵ2],
M=[γ1iΔ1iκiκγ2iΔ2].
M=TΛT1,
Λ=[ω+00ω],T=[τ+τ11],
ω±=γiΔ+iκ2+(Δ+iγ+)2,τ±=Δ+iγ+κ±1+(Δ+iγ+κ)2,
[α+α]=T1[α1α2],
[α˙+α˙]=[ω+00ω][α+α]igom[(λ+α++λα)(β1+β1*)λ(α++α)(β2+β2*)(λ+α++λα)(β1+β1*)+λ+(α++α)(β2+β2*)]+[μ2γ1exϵ1λ2γ2exϵ2μ2γ1exϵ1+λ+2γ2exϵ2],
λ±=Δ1Δ2+i(γ1+γ2)±Ξ12Ξ1,μ=κΞ1,
Ξ1=4κ2+[Δ1Δ2+i(γ1+γ2)]2.
α±ss=μ[ωigom(β2+β2*)]2γ1exϵ1Ξ2±λ[ωigom(β1+β1*)]2γ2exϵ2Ξ2,
Ξ2=ω+ω+igom(ω+λωλ+)(β1+β1*)+igom(ω+λ++ωλ+)(β2+β2*)gom2(λ+λ)2(β1+β1*)(β2+β2*).
α+ssμ2γ1exϵ1λ2γ2exϵ2ω++igomΞ3+λω+ω2γ2exϵ2(ω+ω)2(β1+β1*)+igomΞ3+μω+ω2γ1exϵ1(ω+ω)2(β2+β2*),αssμ2γ1exϵ1λ+2γ2exϵ2ω+igomΞ4++λ+ω+ω2γ2exϵ2(ω+ω)2(β1+β1*)+igomΞ4μω+ω2γ1exϵ1(ω+ω)2(β2+β2*),
Ξ3±=ω(μ2γ1exϵ1λ2γ2exϵ2)(ω+λωλ±),Ξ4±=ω+(μ2γ1exϵ1λ+2γ2exϵ2)(ω+λωλ±).
α1ss=τ+α+ss+ταss=(σ2e1iκe2)Cigom(σ22e1iκσ2e2)C2(β1+β1*)+igom(κ2e1+iκσ1e2)C2(β2+β2*),α2ss=α+ss+αss=(iκe1+σ1e2)C+igom(iκσ2e1+κ2e2)C2(β1+β1*)igom(iκσ1e1+σ12e2)C2(β2+β2*),
C=1/(κ2+δ2+σ2),ek=2γkexϵk,
σ1=γ1+iΔ1,σ2=γ2+iΔ2,δ=Δ1Δ22+iγ1+γ22,σ=γ1+γ22+iΔ1+Δ22.
[β˙1β˙2]=[Γm1i(Ω1+δΩ1)κmechκmechΓm2i(Ω2+δΩ2)]×[β1β2][iη1iη2],
δΩk=4gom2[(κ2Δ1Δ2)Δ3kΔkγ3k2]f2×[γ3k2γkexϵk2+(Δ3kγkexϵkκγ3kexϵ3k)2],κmech=4gom2κ{κγ1exϵ12[Δ2(κ2Δ1Δ22γ1γ2)+Δ1γ22]f2+κγ2exϵ22[Δ1(κ2Δ1Δ2)Δ2γ12]f2γ1exγ2exϵ1ϵ2f2[(κ2γ1γ2)2Δ12Δ22(Δ12γ22Δ22γ12)]},ηk=gom[γ3k2ϵk2+(Δ3kϵkκϵ3k)2]f,
δΩ1=δΩ2gom2Δ(γ2+κ2)2γϵ2[(κ2γ2)2+γ2Δ2]2.
κmech4gom2Δκ2γ3ϵ2[(κ2γ2)2+γ2Δ2]2,
{D+[Γm1+i(Ω1+δΩ1)]}β1+κmechβ2=iη1,κmechβ1+{D+[Γm2+i(Ω2+δΩ2)]}β2=iη2,
λ2+[Γm1+Γm2+i(Ω1+δΩ1+Ω2+δΩ2)]λ+[Γm1+i(Ω1+δΩ1)][Γm2+i(Ω2+δΩ2)]κmech2=0.
λ±=ΓmiΩAve+±iΩAve2κmech2,
ΩAve±=Ω1+δΩ1±(Ω2+δΩ2)2.
λ+=Γmi(2Ω˜2δΩcoup),λ=Γmi(2Ω˜1+δΩcoup),
δΩcoup=Ω˜2Ω˜1(Ω˜2Ω˜1)2κmech2
Ω1,eff=Ω1+δΩ1+δΩcoup,Ω2,eff=Ω2+δΩ2δΩcoup,
Hint=goma1a1(b1+b1)+goma2a2(b2+b2),
geffgom2[γ2(κ2γ2)2+γ2Δ2+1].
α˙k=(γkiΔk)αkiκα3kigomαk(βk+βk*)+2γkexϵk,β˙k=(Γmk+iΩk)βkigom|αk|2.
δΩ1=δΩ2gom2Δϵ2[(κ2+γ1γ2)+Δ+2]2,κmech2δΩ1κ/Δ,
δΩk2gom2Δ(κ2+γ3k2)2ϵ2[(κ2γ1γ2)2+(γ1+γ2)2Δ2/4+Γ2Δ+2]2,κmech2δΩ2κ2γ1γ2κ2+γ12.
ξ1=x1,ξ2=ξ˙1,ξ3=x2,ξ4=ξ˙3;
[ξ˙1(t)ξ˙2(t)ξ˙3(t)ξ˙4(t)]=[0100Ω˜122Γmκmech00001κmech0Ω˜222Γm][ξ1(t)ξ2(t)ξ3(t)ξ4(t)]+[Γ1Γ2Γ3Γ4]=A[ξ1(t)ξ2(t)ξ3(t)ξ4(t)]+[Γ1Γ2Γ3Γ4],
ξi(t)=k=14Gik(t)zk+k=140tGik(t)Γk(tt)dt,
G(t)=eAtIAt[1t00Ω˜12t12Γmtκmecht0001tκmecht0Ω˜22t12Γmt];
ξ1(t)=G11z1+G12z2+G13z3+G14z4+0tG11(t)Γ1(tt)+G12(t)Γ2(tt)+G13(t)Γ3(tt)+G14(t)Γ4(tt)dt=z1+tz2+0ttΓ2(tt)dt.
ξ2(t)=(κmechz3+Ω˜12z1)t+(12Γmt)z2+0t(12Γmt)Γ2(tt)dt,ξ3(t)=z3+z4t+0ttΓ4(tt)dt,ξ4(t)=(κmechz1+Ω˜22z3)t+(12Γmt)z4+0t(12Γmt)Γ4(tt)dt.
Rij(τ,t)=ξi(t+τ)ξj(t),
Rij(τ,t)=k=14Gik(τ)ξk(t)ξj(t),0τ.
R13(τ,t)=(z1+tz2)(z3+z4t)+q3t3+τ[(κmechz3+Ω˜12z1)(z3+z4t)t+(12Γmt)(z3+z4t)z2+q(12t223Γmt3)],R11(0,t)=(z1+z2t)2+q3t3,R33(0,t)=(z3+z4t)2+q3t3.
R(τ,t)=|R13(τ,t)|R11(0,t)R33(0,t)=|12Ω˜12τt+qκmechΩ˜122τt2+qκmechΩ˜123τt3|1+q3κmech2t31+q3Ω˜14t312Ω˜12τt+q2κmechΩ˜12τt2+q3κmechΩ˜12τt312Ω˜12τt+2ΓmkTκmechΩ˜12τt2+43ΓmkTκmechΩ˜12τt3.