Long-distance synchronization of unidirectionally cascaded optomechanical systems

Synchronization is of great scientific interest due to the abundant applications in a wide range of systems. We propose a scheme to achieve the controllable long-distance synchronization of two dissimilar optomechanical systems, which are unidirectionally coupled through a fiber with light. Synchronization, unsynchronization, and the dependence of the synchronization on driving laser strength and intrinsic frequency mismatch are studied based on the numerical simulation. Taking the fiber attenuation into account, it's shown that two mechanical resonators can be synchronized over a distance of tens of kilometers. In addition, we also analyze the unidirectional synchronization of three optomechanical systems, demonstrating the scalability of our scheme.


I. INTRODUCTION
Synchronization is a universal phenomenon in nature, where oscillators with different intrinsic frequencies can adjust their rhythms to oscillate in unison [1,2]. In 1660s, Huygens observed the synchronization of two pendulum clocks hanging on a same wall [3]. Since then, synchronization has been observed in a wide range of systems. For example, the coordination of neurons [4] and the regular flash of glowworms colonies [5]. Synchronization is of importance for both fundamental research and practical applications, since it has the capacity to improve the precision [6] of frequency sources built from (electro)mechanical oscillators in producing oscillating signals, which plays a critical role in time-keeping [7], sensing [8] and communication [9].
However, those OMSs are coupled through local optical coupling between cavities, while the greatest advantage of the light that can propagate over very long distance is overlooked. Very recently, a long-distance master-slave * clzou321@ustc.edu.cn † xbz@ustc.edu.cn frequency locking has been realized between two OMSs [29], while the light output from one OMS is converted to radio frequency (RF) signal and the other OMS is injection locked by using an electro-optic modulator (EOM) to modulate the input laser. Extra elements required in this scheme, such as detectors and amplifiers will introduce noises to such system and may limit the stability of the system. In this paper, we present a scheme to realize synchronization of cascaded OMSs, where two OMSs are coupled through light propagating unidirectionally in fiber, no extra detection of light is required. Through numerical simulation, we observe the synchronization phenomenon and study the influence of different systemic and external driving parameters on synchronization. In practical applications in long distance synchronization, we take the fiber attenuation into account, and confirm the synchronization is possible for two OMSs over tens of kilometers. Last but not least, we expand synchronization of two OMSs into synchronization of three OMSs, which verifies the feasibility of unidirectional synchronization of an OMSs array.

II. MODEL
The unidirectionally cascaded synchronization scheme consists of two toroid optical microcavities [30] with small mechanical frequency mismatch. Both toroids are cascaded coupled with the optical fiber, as shown in Fig. 1. The input laser in the fiber is coupled to the traveling optical whispering gallery modes in the former toroid, and the transmitted light is coupled to the following toroid. Each toroid also supports low loss mechanical breath vibration mode [31], thus enables optomechanical coupling. In our model, it's assumed that there is no laser input in the reversal direction, so the optical coupling between cascaded toroids are unidirectional. We would expect that light could carry the vibration information from the first optomechanical system (OMS-1) to the second op-tomechanical system (OMS-2), and thus enable the unidirectional synchronization. The Hamiltonian of the individual OMS-j (j = 1, 2) is where a † j (b † j ) and a j (b j ) are the optical (mechanical) creation and annihilation operators, frequencies of optical and mechanical mode are denoted as ω cj and ω mj respectively. The last term describes the dispersive coupling of optical mode and mechanical mode, where g j is the vacuum optomechanical coupling rate.
The dynamics of the unidirectionally cascaded OMSs are determined by the quantum Langevin equations, where O is an arbitrary system operator, N and H diss represent the environment noises and the system dissipation respectively. In the semiclassical cases, the mean values of the environment noises vanish, thus the equations of motion are as follows, where j = 1, 2. κ j and κ exj are the total and external optical decay rates, respectively. γ mj is the mechanical damping rate. a where η 12 = √ η P and η P is the power transmittance, a (1) out (t − τ ) represents the output field of OMS-1, τ is the required time for light transmitting from OMS-1 to OMS-2. In the case of unidirectionally cascaded systems, only one direction for transmission is allowed. Thus, without loss of universality, we let τ → 0 + . Based on the input and output theory of optical cavities [35], Assuming a (1) where E in and ω L represent the strength and frequency of the driving optical field, respectively. In the rotating frame with the driving frequency ω L , defineã j = a j e iωLt (j = 1, 2), then based on Eq. (2), the optical and mechanical modes satisfẏ where In addition, the equations of motion can also be derived consistently from the master equation [36], indicating the time evolution of density matrix ρ, where And the coupling term in the master equation consists of a damping term L [a 1 + a 2 ] ρ and a commutator 1 2 a † 1 a 2 − a † 2 a 1 , ρ , which indicates the system's unidirectionality.
From equations of motion [Eqs. (5) and (6)], the output of OMS-1 drives the optical mode of OMS-2. In contrast, the output of OMS-2 has no effects on OMS-1. Due to the nonlinear interaction between optical mode and mechanical mode, such as g 1ã1 b 1 + b † 1 in Eq. (5), the output optical field from OMS-1 can modify the behavior of the mechanical resonator in OMS-2, and may lead to the synchronization. The dynamics of the system is significantly different from previously studied bidirectionally coupled OMSs, where the mutual coupling can induce the synchronization.

III. UNIDIRECTIONAL SYNCHRONIZATION
Since the nonlinear optomechanical interaction is crucial in the synchronization, we don't apply linear approximations to solve the equations of motion. The full dynamics of unidirectionally cascaded systems are simulated for long evolution time numerically. For the convenient to illustrate the synchronization, the optical and mechanical operators are re-written as where j = 1, 2. And the corresponding equations for quadratures of optical fields Q j , P j and mechanical displacement q j and momentum p j reaḋ where G j = √ 2g j . The numerical simulation is performed using the four-order Runge-Kutta algorithm. In the simulation, we choose realistic values of the parameters [17,23] and normalize them by ω m1 : ω m1 = 1, ω m2 = 1.005, i.e., the intrinsic frequency of OMS-2 differs from that of OMS-1 with a mismatch of 5 ω m1 . ∆ 1 = −ω m1 , ∆ 2 = −ω m2 , i.e., the driving laser is blue detuned, which guarantees that OMS-1 will evolve into self-sustained oscillation as long as the driving strength is strong enough. The other parameters are In addition, the time scale in the simulation becomes dimensionless and changes from t into t ′ = ω m1 t due to the normalization.
Firstly, we study the general properties of lossless cascade coupling between two OMSs. Figure 2 shows typical behaviors of the OMSs for different parameters. Under the effective driving of E = 64, the dynamical evolution of mechanical displacement for two OMSs are shown in Fig. 2(a), and the corresponding power spectrum density (PSD) and phase diagram are shown in Fig. 2(b) and Fig. 2(c), respectively. Although the intrinsic mechanical frequencies are different by 5 , the eventual oscillation frequencies are the same ω ′ m1 = ω ′ m2 = 0.990932. The regular orbit in phase diagram confirms that the oscillation periods are exactly the same and their phase difference is constant. To exclude the possible coincidence that the self-oscillation frequencies of two OMSs under external optical driving are the same, we also plot the PSD for OMSs individually driven by the external laser in Fig. 2(d)   not affected by the OMS-2, and the OMS-2 is synchronized to OMS-1 under the unidirectional optical coupling. With a further increase in the strength of the laser driving the OMSs, the two OMSs are not guaranteed to be synchronized under the unidirectional coupling. As shown in Figs. 2(e)-2(h), the OMSs are unsynchronized for E = 100. From the PSD, the OMS-1 is still unaffected by the OMS-2, just shows a single peak self-oscillation behavior. However, the PSD of OMS-2 [ Fig. 2(f)] shows multiple peaks when driven by the output from OMS-1. The frequency locations of those peaks show equal distances. This can be interpreted as the dynamics of OMS-2 can still be greatly affected by OMS-1 for large laser driving, but nonlinear effect generates the frequency mixing of two systems instead of synchronization, which is a typical feature of the well-known nonlinear periodic pulling [37][38][39].
It is quite straightforward that there is also a threshold for nonlinear optomechanical interaction to make synchronization happen. The above results also indicate that the synchronization effect can only dominate the other nonlinear effects in certain driving laser amplitudes. For example, very strong nonlinear effect will induce multistable and even chaotic dynamics. Therefore, we further study the final frequencies ω ′ m1 , ω ′ m2 as functions of the effective driving strength E. For each E, we try 10 sets of different random initial values of the system to test the sensitivity of the synchronization to initial conditions. In Fig. 3, the PSD of q 1 and the PSD of q 2 but shifted in respect to the spectrum of q 1 are plotted. The results reveals different dynamical regimes for unidirectionally coupled OMSs: (1) Weak nonlinear effect, E ∈ [10,37]. Below the threshold of about E ≈ 37, the two OMSs are unsynchronized ω ′ m2 = ω ′ m1 . However, the OMS-2 are affected by the mechanical oscillation in OMS-1, thus a series of sidebands appear in the PSD of q 2 . (2) Synchronization, E ∈ [38,96]. For moderate driving, the two OMSs are synchronized with a sole peak in the PSD of q 2 and ω ′ m2 = ω ′ m1 . (3) Multi-stable and chaotic regime, E ∈ [97, 160]. For very strong driving, the two OMSs are unsynchronized. There are multiple possible selfoscillation frequencies of OMS-1. For certain OMS-1 oscillation frequency, the OMS-2 can still be synchronized. While, for other frequencies, the OMS-2 exhibits very complex dynamics, including synchronization, frequency mixing and multi-stable dynamics simultaneously. Actually, due to the unidirectionality, OMS-1 is independent from OMS-2 and thus can be fully theoretically solved using the single OMS theory [40] and the output field is modulated by the mechanical vibration. The observed synchronization and periodic pulling phenomena of OMS-2 originate from the modulated laser driving on OMS-2. Similar effects have been demonstrated with the injection-locking [41][42][43] of an OMS [29,44,45], where the input laser of the OMS is partially modulated by a single tone RF signal using an electro-optic modulator.
Previous studies show that synchronization occurs only when the driving RF frequency is very close to the intrinsic oscillation frequency [38].

IV. LONG DISTANCE UNIDIRECTIONAL SYNCHRONIZATION WITH FIBER-LOSS
The unidirectional coupling is very potential for future long distance synchronization, since the OMSs are directly coupled through the optical connections, without extra optical-to-electronic or reversal conversion processes. In addition, the unidirectional coupling also greatly reduces the complexity of experiments. To testify the potential for long distance synchronization, we take the practical fiber attenuation loss into our model. Take 1550nm light as an example, the propagation loss rate is α = 0.2 dB/km, the power transmittance η P = 10 −αL/10 .  First, take L = 4.6 km, i.e., η 12 = √ η P = 0.9, synchronization region of Fig. 5(a)]. Compared to the result without fiber loss η P = 1.0 in Fig. 4, the parameter region has shrunk. Then, we explore whether the systems are synchronized or not for L varying from 0 to 80km, while fixing the intrinsic mechanical frequencies ω m1 = 1, ω m2 = 1.005. As plotted in Fig. 5(b), we can see for each E there exists a critical distance L cri , over which the state of two OMSs changes from synchronization into unsynchronization. The critical distance for E = 40 and E = 64 are as long as 1.3 km and 16.7 km, respectively, which verifies the capability of our scheme to realize longdistance unidirectional synchronization.

VI. CONCLUSION
We have demonstrated the synchronization of optomechanical systems by all-optical method, where the systems are coupled through light propagating unidirectionally in the fiber. For two OMSs with fixed mechanical frequency mismatch, synchronization can be tuned on or off through tuning the optical driving strength. For a fixed driving strength, there exists a region of mechanical frequency mismatch that allows for the synchronization. And in the practical cases, the synchronization can still be achieved for distance over 10 km, while the synchronization region shrinks due to the light attenuates when travel over long distances. Unidirectional synchronization of three OMSs is also obtained, as well. The alloptical feature, high controllability, wide synchronization region, long synchronization distance, and novel scalability of our scheme are appealing and can be useful for many applications, such as the construction of complex synchronization OMSs networks [28]. We expected that the scheme also works for other optomechanical interactions, such as quadratic [46] , dissipative [47,48] and Brillouin [49,50] optomechanical interactions.