Long-distance synchronization of unidirectionally cascaded optomechanical systems
Tan Li, Tian-Yi Bao, Yan-Lei Zhang, Chang-Ling Zou, Xu-Bo Zou, Guang-Can Guo
aa r X i v : . [ phy s i c s . op ti c s ] D ec Long-distance synchronization of unidirectionally cascaded optomechanical systems
Tan Li, , , , Tian-Yi Bao, , Yan-Lei Zhang, , , Chang-Ling Zou, , , ∗ Xu-Bo Zou, , , † and Guang-Can Guo, , Zhengzhou Information Science and Technology Institute, Zhengzhou, Henan 450004, China Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China and Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, China
Synchronization is of great scientific interest due to the abundant applications in a wide rangeof systems. We propose a scheme to achieve the controllable long-distance synchronization of twodissimilar optomechanical systems, which are unidirectionally coupled through a fiber with light.Synchronization, unsynchronization, and the dependence of the synchronization on driving laserstrength and intrinsic frequency mismatch are studied based on the numerical simulation. Takingthe fiber attenuation into account, it’s shown that two mechanical resonators can be synchronizedover a distance of tens of kilometers. In addition, we also analyze the unidirectional synchronizationof three optomechanical systems, demonstrating the scalability of our scheme.
PACS numbers: 05.45.Xt, 42.50.Wk, 42.82.Et, 07.10.Cm
I. INTRODUCTION
Synchronization is a universal phenomenon in nature,where oscillators with different intrinsic frequencies canadjust their rhythms to oscillate in unison [1, 2]. In 1660s,Huygens observed the synchronization of two pendulumclocks hanging on a same wall [3]. Since then, synchro-nization has been observed in a wide range of systems.For example, the coordination of neurons [4] and the reg-ular flash of glowworms colonies [5]. Synchronization isof importance for both fundamental research and prac-tical applications, since it has the capacity to improvethe precision [6] of frequency sources built from (elec-tro)mechanical oscillators in producing oscillating sig-nals, which plays a critical role in time-keeping [7], sens-ing [8] and communication [9].Synchronization has been demonstrated in many sys-tems, such as Josephson junctions [10, 11], micro- andnano- electromechanical systems [12–15], ensembles ofatoms [16]. Optomechanical system (OMS) [17–19] is oneof such platforms for synchronization research [20–24],and holds great potential for applications due to the eas-ily fabrication, high quality factor of optical resonatorsand strong optomechanical coupling. The synchroniza-tion of OMSs have been predicted theoretically [25] anddemonstrated in experiments [26–28]. For example, intwo silicon nitride microdisks, spaced apart by 400nm,two mechanical modes are synchronized by the couplingof two optical modes [26]. Two spatially separated 80micrometers nanomechanical oscillators are also synchro-nized through coupling to a same racetrack cavity [27].However, those OMSs are coupled through local opticalcoupling between cavities, while the greatest advantageof the light that can propagate over very long distance isoverlooked. Very recently, a long-distance master-slave ∗ [email protected] † [email protected] frequency locking has been realized between two OMSs[29], while the light output from one OMS is converted toradio frequency (RF) signal and the other OMS is injec-tion locked by using an electro-optic modulator (EOM)to modulate the input laser. Extra elements required inthis scheme, such as detectors and amplifiers will intro-duce noises to such system and may limit the stability ofthe system.In this paper, we present a scheme to realize synchro-nization of cascaded OMSs, where two OMSs are coupledthrough light propagating unidirectionally in fiber, noextra detection of light is required. Through numericalsimulation, we observe the synchronization phenomenonand study the influence of different systemic and externaldriving parameters on synchronization. In practical ap-plications in long distance synchronization, we take thefiber attenuation into account, and confirm the synchro-nization is possible for two OMSs over tens of kilometers.Last but not least, we expand synchronization of twoOMSs into synchronization of three OMSs, which veri-fies the feasibility of unidirectional synchronization of anOMSs array. II. MODEL
The unidirectionally cascaded synchronization schemeconsists of two toroid optical microcavities [30] with smallmechanical frequency mismatch. Both toroids are cas-caded coupled with the optical fiber, as shown in Fig. 1.The input laser in the fiber is coupled to the traveling op-tical whispering gallery modes in the former toroid, andthe transmitted light is coupled to the following toroid.Each toroid also supports low loss mechanical breath vi-bration mode [31], thus enables optomechanical coupling.In our model, it’s assumed that there is no laser inputin the reversal direction, so the optical coupling betweencascaded toroids are unidirectional. We would expectthat light could carry the vibration information from thefirst optomechanical system (OMS-1) to the second op-tomechanical system (OMS-2), and thus enable the uni-directional synchronization. a a b b (cid:11) (cid:12) a (cid:11) (cid:12) a (cid:11) (cid:12) a (cid:11) (cid:12) aL Figure 1. (Color online) Schematic setup of the unidirection-ally cascaded systems of two toroid optical microcavities, cou-pled through a unidirectional fiber with light. The distancebetween the two OMSs is denoted as L . The Hamiltonian of the individual OMS- j ( j = 1 ,
2) is H j = ~ ω cj a † j a j + ~ ω mj b † j b j − ~ g j a † j a j (cid:16) b j + b † j (cid:17) , (1)where a † j ( b † j ) and a j ( b j ) are the optical (mechanical) cre-ation and annihilation operators, frequencies of opticaland mechanical mode are denoted as ω cj and ω mj re-spectively. The last term describes the dispersive cou-pling of optical mode and mechanical mode, where g j isthe vacuum optomechanical coupling rate.The dynamics of the unidirectionally cascaded OMSsare determined by the quantum Langevin equations, ∂O∂t = i ~ [ O, H ] + N − H diss , where O is an arbitrarysystem operator, N and H diss represent the environmentnoises and the system dissipation respectively. In thesemiclassical cases, the mean values of the environmentnoises vanish, thus the equations of motion are as follows,˙ a j = 1 i ~ [ a j , H j ] − κ j a j + √ κ exj a ( j ) in , ˙ b j = 1 i ~ [ b j , H j ] − γ mj b j , (2)where j = 1 , κ j and κ exj are the total and externaloptical decay rates, respectively. γ mj is the mechanicaldamping rate. a ( j ) in represents the injected driving field.Based on the properties of cascaded systems [32–34], a (2) in ( t ) = η a (1) out ( t − τ ) , (3)where η = √ η P and η P is the power transmittance, a (1) out ( t − τ ) represents the output field of OMS-1, τ is therequired time for light transmitting from OMS-1 to OMS-2. In the case of unidirectionally cascaded systems, onlyone direction for transmission is allowed. Thus, without loss of universality, we let τ → + . Based on the inputand output theory of optical cavities [35], a (1) out ( t ) = a (1) in ( t ) − √ κ ex a . (4)Assuming a (1) in ( t ) = E in e − iω L t , where E in and ω L rep-resent the strength and frequency of the driving opticalfield, respectively. In the rotating frame with the drivingfrequency ω L , define ˜ a j = a j e iω L t ( j = 1 , a = − ( i∆ + κ a + ig ˜ a (cid:16) b + b † (cid:17) + E, (5)˙˜ a = − ( i∆ + κ a + ig ˜ a (cid:16) b + b † (cid:17) + η √ κ ex ( E/ √ κ ex − √ κ ex ˜ a ) , (6)˙ b j = − iω mj b j + ig j ˜ a † j ˜ a j − γ mj b j , j = 1 , , (7)where ∆ j = ω cj − ω L ( j = 1 ,
2) is the driving field de-tuning. E = √ κ ex E in is the effective optical drivingstrength of OMS-1.In addition, the equations of motion can also be derivedconsistently from the master equation [36], indicating thetime evolution of density matrix ρ ,˙ ρ = 1 i ~ [ H + H , ρ ] + κ L [ a ] ρ + κ L [ a ] ρ + γ m L [ b ] ρ + γ m L [ a ] ρ + √ κ ex κ ex (cid:26) L [ a + a ] ρ + 12 h a † a − a † a , ρ i(cid:27) + h a (1) in (cid:16) √ κ ex a † + √ κ ex a † (cid:17) − h.c, ρ i , (8)where L [ o ] ρ = oρo † − (cid:0) o † oρ + ρo † o (cid:1) is the Lindblad su-peroperator. And the coupling term in the master equa-tion consists of a damping term L [ a + a ] ρ and a com-mutator h a † a − a † a , ρ i , which indicates the system’sunidirectionality.From equations of motion [Eqs. (5) and (6)], the out-put of OMS-1 drives the optical mode of OMS-2. Incontrast, the output of OMS-2 has no effects on OMS-1.Due to the nonlinear interaction between optical modeand mechanical mode, such as g ˜ a (cid:16) b + b † (cid:17) in Eq. (5),the output optical field from OMS-1 can modify the be-havior of the mechanical resonator in OMS-2, and maylead to the synchronization. The dynamics of the systemis significantly different from previously studied bidirec-tionally coupled OMSs, where the mutual coupling caninduce the synchronization. III. UNIDIRECTIONAL SYNCHRONIZATION
Since the nonlinear optomechanical interaction is cru-cial in the synchronization, we don’t apply linear ap-proximations to solve the equations of motion. The fulldynamics of unidirectionally cascaded systems are simu-lated for long evolution time numerically. For the con-venient to illustrate the synchronization, the optical andmechanical operators are re-written as Q j = (cid:16) ˜ a j + ˜ a † j (cid:17) / , P j = − i (cid:16) ˜ a j − ˜ a † j (cid:17) / ,q j = (cid:16) b j + b † j (cid:17) / √ , p j = − i (cid:16) b j − b † j (cid:17) / √ , (9)where j = 1 ,
2. And the corresponding equations forquadratures of optical fields Q j , P j and mechanical dis-placement q j and momentum p j read˙ Q = ( ∆ − G q ) P − κ Q + E, ˙ P = − ( ∆ − G q ) Q − κ P , ˙ Q = ( ∆ − G q ) P − κ Q + √ κ ex ( E/ √ κ ex − √ κ ex Q ) , ˙ P = − ( ∆ − G q ) Q − κ P − √ κ ex κ ex P , ˙ q j = ω mj p j , ˙ p j = − ω mj q j − γ mj p j + G j (cid:0) ˜ a jr + ˜ a ji (cid:1) , (10)where G j = √ g j . The numerical simulation is per-formed using the four-order Runge-Kutta algorithm. Inthe simulation, we choose realistic values of the param-eters [17, 23] and normalize them by ω m : ω m = 1 ,ω m = 1 . , i.e., the intrinsic frequency of OMS-2 dif-fers from that of OMS-1 with a mismatch of 5 h ω m . ∆ = − ω m , ∆ = − ω m , i.e., the driving laser is bluedetuned, which guarantees that OMS-1 will evolve intoself-sustained oscillation as long as the driving strengthis strong enough. The other parameters are G = G =4 × − , κ = κ = 0 . , κ ex = κ , κ ex = κ , γ m = γ m = 5 × − . In addition, the time scale inthe simulation becomes dimensionless and changes from t into t ′ = ω m t due to the normalization.Firstly, we study the general properties of lossless cas-cade coupling between two OMSs. Figure 2 shows typicalbehaviors of the OMSs for different parameters. Underthe effective driving of E = 64, the dynamical evolu-tion of mechanical displacement for two OMSs are shownin Fig. 2(a), and the corresponding power spectrum den-sity (PSD) and phase diagram are shown in Fig. 2(b) andFig. 2(c), respectively. Although the intrinsic mechanicalfrequencies are different by 5 h , the eventual oscillationfrequencies are the same ω ′ m = ω ′ m = 0 . ω ′ m = 0 . ω ′ m = 0 . h similar to that of intrinsic frequencies. The re-sults confirm that ω ′ m are exactly the same as ω m and Time t ′ × -100001000 q , q q q Time t ′ × -200002000 q , q q q Frequency -50050100 PS D ( d B m ) q q Frequency -50050100 PS D ( d B m ) q q -1000 0 1000 q -100001000 q -2000 0 2000 q -200002000 q Frequency -50050100 PS D ( d B m ) q q Frequency -50050100 PS D ( d B m ) q q (a)(c)(d) (e)(b) (f)(g)(h) Figure 2. (Color online) Numerical solutions of equations ofmotion in the unidirectionally cascaded two OMSs scheme,for the parameters: ω m = 1, ω m = 1 . ∆ = − ω m , ∆ = − ω m , G = G = 4 × − , κ = κ = 0 . , κ ex = κ ex = 0 . γ m = γ m = 5 × − . (a) and (e) Dynamicalevolution. (b) and (f) Power spectrum density (PSD) of thedisplacement operators q and q . (c) and (g) Phase diagramof q , q . (d) and (h) Displacement PSD of each single OMSdriven by a constant amplitude optical field E . The left andright columns correspond to E = 64 and E = 100. not affected by the OMS-2, and the OMS-2 is synchro-nized to OMS-1 under the unidirectional optical coupling.With a further increase in the strength of the laserdriving the OMSs, the two OMSs are not guaranteed tobe synchronized under the unidirectional coupling. Asshown in Figs. 2(e)-2(h), the OMSs are unsynchronizedfor E = 100. From the PSD, the OMS-1 is still unaffectedby the OMS-2, just shows a single peak self-oscillationbehavior. However, the PSD of OMS-2 [Fig. 2(f)] showsmultiple peaks when driven by the output from OMS-1. The frequency locations of those peaks show equaldistances. This can be interpreted as the dynamics ofOMS-2 can still be greatly affected by OMS-1 for largelaser driving, but nonlinear effect generates the frequencymixing of two systems instead of synchronization, whichis a typical feature of the well-known nonlinear periodicpulling [37–39].It is quite straightforward that there is also a thresh-old for nonlinear optomechanical interaction to make syn-chronization happen. The above results also indicate thatthe synchronization effect can only dominate the othernonlinear effects in certain driving laser amplitudes. Forexample, very strong nonlinear effect will induce multi-stable and even chaotic dynamics. Therefore, we furtherstudy the final frequencies ω ′ m , ω ′ m as functions of theeffective driving strength E . For each E , we try 10 setsof different random initial values of the system to test thesensitivity of the synchronization to initial conditions.In Fig. 3, the PSD of q and the PSD of q but shiftedin respect to the spectrum of q are plotted. The resultsreveals different dynamical regimes for unidirectionallycoupled OMSs: (1) Weak nonlinear effect, E ∈ [10 , E ≈
37, the two OMSs areunsynchronized ω ′ m = ω ′ m . However, the OMS-2 areaffected by the mechanical oscillation in OMS-1, thus aseries of sidebands appear in the PSD of q . (2) Synchro-nization, E ∈ [38 , q and ω ′ m = ω ′ m . (3) Multi-stable and chaotic regime, E ∈ [97 , Figure 3. (Color online) (a) The PSD of q as a function ofthe effective driving strength E , with a sole frequency peakdenoted as ω ′ m . (b) The PSD of q relative to ω ′ m as afunction of the effective driving strength E . The maximumfrequency component in the PSD of q without the frequencyshift is denoted as ω ′ m . The color scaled in the color barindicates the power values in the PSD. The other simulationparameters are the same as those in Fig. 2. Actually, due to the unidirectionality, OMS-1 is inde-pendent from OMS-2 and thus can be fully theoreticallysolved using the single OMS theory [40] and the output field is modulated by the mechanical vibration. The ob-served synchronization and periodic pulling phenomenaof OMS-2 originate from the modulated laser driving onOMS-2. Similar effects have been demonstrated with theinjection-locking [41–43] of an OMS [29, 44, 45], wherethe input laser of the OMS is partially modulated by asingle tone RF signal using an electro-optic modulator.Previous studies show that synchronization occursonly when the driving RF frequency is very closeto the intrinsic oscillation frequency [38]. Inspir-ited by this, the final relative frequency difference( ω ′ m − ω ′ m ) /ω ′ m as a function of the intrinsic fre-quency mismatch ( ω m − ω m ) /ω m is plotted in Fig. 4.When E = 64 ( E = 40), there is a synchroniza-tion region of ω m ∈ [1 − . h , . h ] ω m ( ω m ∈ [1 − h , . h ] ω m ), represented by the red line (theblue line). We find that, the width of synchronizationregion is similar to the mechanical damping rate 5 h [45], and the increase of driving strength does enlargethe width of synchronization region, by comparing theresults of E = 64 and that of E = 40. -0.01 0 0.01 ( ω m − ω m ) /ω m -0.0100.01 ( ω ′ m − ω ′ m ) / ω ′ m E = 40 E = 64none coupling Figure 4. (Color online) The final relative frequency differ-ence between the maximum frequency components of twoOMSs ( ω ′ m − ω ′ m ) /ω ′ m vs the relative intrinsic frequencymismatch ( ω m − ω m ) /ω m . Blue solid line marked with’x’: E = 40. Red solid line marked with ’o’: E = 64. Blacksolid line: uncoupled free-running case. IV. LONG DISTANCE UNIDIRECTIONALSYNCHRONIZATION WITH FIBER-LOSS
The unidirectional coupling is very potential for fu-ture long distance synchronization, since the OMSs aredirectly coupled through the optical connections, with-out extra optical-to-electronic or reversal conversion pro-cesses. In addition, the unidirectional coupling alsogreatly reduces the complexity of experiments. To testifythe potential for long distance synchronization, we takethe practical fiber attenuation loss into our model. Take1550nm light as an example, the propagation loss rate is α = 0 . / km, the power transmittance η P = 10 − αL/ . -0.01 0 0.01 ( ω m − ω m ) /ω m -0.0100.01 ( ω ′ m − ω ′ m ) / ω ′ m (a) E = 40 E = 64none coupling Distance L (/km) ω ′ m , ω ′ m (b) ω ′ m with E = 40 ω ′ m with E = 64 ω ′ m with E = 40 ω ′ m with E = 64 Figure 5. (Color online) Considering fiber-loss, (a) the fi-nal relative frequency difference between the maximum fre-quency components of the two OMSs ( ω ′ m − ω ′ m ) /ω ′ m vsthe relative intrinsic frequency mismatch ( ω m − ω m ) /ω m with L = 4 . η = 0 .
9. The cases corresponding to thelines are the same as those in Fig. 4. (b) The final maximumfrequency components of the two OMSs ω ′ m , ω ′ m vs distance L . Blue solid line marked with ’x’: ω ′ m with E = 40. Redsolid line marked with ’o’: ω ′ m with E = 64. Blue dash-dotted line: ω ′ m with E = 40. Red dashed line: ω ′ m with E = 64. First, take L = 4 . η = √ η P = 0 .
9, syn-chronization region of ω m ∈ [1 − . h , . h ] ω m ( ω m ∈ [1 − . h , . h ] ω m ) is revealed for E = 64( E = 40) [Fig. 5(a)]. Compared to the result withoutfiber loss η P = 1 . L varying from 0 to 80km, whilefixing the intrinsic mechanical frequencies ω m = 1 ,ω m = 1 . E there exists a critical distance L cri , over which the state of two OMSs changes from synchronization intounsynchronization. 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 long-distance unidirectional synchronization. V. UNIDIRECTIONAL SYNCHRONIZATIONOF THREE OMSS
Now, we further study the generalized unidirectionalsynchronization of a cascaded OMSs array. From the re-sults of two OMSs, the synchronization is possible foradditional OMSs following OMS-2, as long as the driv-ing laser intensity on them is moderate and contains thecomponents of modulation from the mechanical vibra-tions. As an example, the unidirectional synchronizationscheme of three dissimilar OMSs (microtoroids) with dif-ferent intrinsic frequencies are cascaded using a unidirec-tional fiber, as shown in Fig. 6. a a a b b b (cid:11) (cid:12) a (cid:11) (cid:12) a (cid:11) (cid:12) a (cid:11) (cid:12) a (cid:11) (cid:12) a (cid:11) (cid:12) a L L Figure 6. (Color online) Schematic setup of the unidirection-ally cascaded synchronization scheme consists of three OMSs.The distance between the first (last) two OMSs is denoted as L ( L ). Let η = a (3) in /a (2) out , η = a (2) in /a (1) out , the set of equa-tions of motion can be obtained, where ˙˜ a , ˙˜ a , ˙ b and ˙ b are the same as Eqs.(5,6,7) due to the unidirectionality,and ˙˜ a and ˙ b are in the following form,˙˜ a = − i (cid:16) ∆ + κ (cid:17) ˜ a + ig ˜ a (cid:16) b + b † (cid:17) + η η √ κ ex ( E/ √ κ ex − √ κ ex ˜ a ) − η √ κ ex κ ex ˜ a , ˙ b = − iω m b + ig ˜ a † ˜ a − γ m b . (11)Following the similar procedure of numerical simula-tion for two OMSs, the dynamics of the three OMSsare solved. Shown in Fig. 7(a) (Fig. 7(b)) the PSDs of q , q and q under a set of parameters: ω m = 0 . ω m = 1 .
005 ( ω m = 1 . ω m = 1 . E = 64, ω m = 1, η = η = 1, ∆ = ∆ = ∆ = − ω m , G = G = G = 0 . , κ = κ = κ = 0 . ,κ ex = κ ex = κ ex = 0 . γ m = γ m = γ m = 0 . Frequency PS D ( d B m ) (a) q q q Frequency -50050100 PS D ( d B m ) (b) q q q -0.02 -0.01 0 0.01 0.02 ( ω m − ω m ) /ω m -0.02-0.0100.010.02 T he f i na l r e l a t i v e f r equen cy (c) ( ω ′ m − ω ′ m ) /ω ′ m ( ω ′ m − ω ′ m ) /ω ′ m none coupling -0.02 -0.01 0 0.01 0.02 ( ω m − ω m ) /ω m -0.02-0.0100.010.02 T he f i na l r e l a t i v e f r equen cy (d) ( ω ′ m − ω ′ m ) /ω ′ m ( ω ′ m − ω ′ m ) /ω ′ m none coupling Figure 7. (Color online) Numerical solutions of the equa-tions of motion in the unidirectionally cascaded three OMSsscheme. (a) and (b) The PSDs of q , q and q . The parame-ters are: (a) ω m = 0 . ω m = 1 . ω m = 1 . ω m = 1 . ω m relative to ω m . The parameters are: (c) ω m = 1, ω m = 0 . ω m = 1, ω m = 1 . In addition, for a fixed effective driving strength E = 64 and ω m = 1, when ω m = 0 . ω m = 1 . ω m ∈ [1 − . h , . h ] ω m ( ω m ∈ [1 − h , . h ] ω m ) can be obtained by traversing ω m ∈ [0 . , . ω m . Thus,the synchronization of three OMSs is available, whichverifies the feasibility of synchronization of an array ofmore than 2 cascaded OMSs. VI. CONCLUSION
We have demonstrated the synchronization of optome-chanical systems by all-optical method, where the sys-tems are coupled through light propagating unidirection-ally in the fiber. For two OMSs with fixed mechanicalfrequency mismatch, synchronization can be tuned on oroff through tuning the optical driving strength. For afixed driving strength, there exists a region of mechani-cal frequency mismatch that allows for the synchroniza-tion. And in the practical cases, the synchronization canstill be achieved for distance over 10 km, while the syn-chronization region shrinks due to the light attenuateswhen travel over long distances. Unidirectional synchro-nization of three OMSs is also obtained, as well. The all-optical feature, high controllability, wide synchronizationregion, long synchronization distance, and novel scalabil-ity of our scheme are appealing and can be useful formany applications, such as the construction of complexsynchronization OMSs networks [28]. We expected thatthe scheme also works for other optomechanical inter-actions, such as quadratic [46] , dissipative [47, 48] andBrillouin [49, 50] optomechanical interactions.
ACKNOWLEDGMENTS
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