Abstract

Synchronization is of great scientific interest due to the abundant applications in a wide range of systems. We propose an all-optical 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, we show that two optomechanical resonators can be unidirectionally synchronized over a distance of tens of kilometers. We also analyze the unidirectional synchronization of three optomechanical systems, demonstrating the scalability of our scheme.

© 2016 Optical Society of America

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References

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2015 (8)

D. Antonio, D. A. Czaplewski, J. R. Guest, D. Lpez, S. I. Arroyo, and D. H. Zanette, “Nonlinearity–induced synchronization enhancement in micromechanical oscillators,” Phys. Rev. Lett. 114, 034103 (2015).
[Crossref]

W. Li, C. Li, and H. Song, “Criterion of quantum synchronization and controllable quantum synchronization based on an optomechanical system,” J. Phys. B 48(3), 035503 (2015).
[Crossref]

K. Shlomi, D. Yuvaraj, I. Baskin, O. Suchoi, R. Winik, and E. Buks, “Synchronization in an optomechanical cavity,” Phys. Rev. E 91(3), 032910 (2015).
[Crossref]

M. Zhang, S. Shah, J. Cardenas, and M. Lipson, “Synchronization and phase noise reduction in micromechanical oscillator arrays coupled through light,” Phys. Rev. Lett. 115, 163902 (2015).
[Crossref] [PubMed]

S. Y. Shah, M. Zhang, R. Rand, and M. Lipson, “Master-slave locking of optomechanical oscillators over a long distance,” Phys. Rev. Lett. 114(11), 113602 (2015).
[Crossref] [PubMed]

D. Lee, M. Underwood, D. Mason, A. Shkarin, S. Hoch, and J. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Commun. 6, 6232 (2015).
[Crossref] [PubMed]

C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

2014 (4)

L. Ying, Y.-C. Lai, and C. Grebogi, “Quantum manifestation of a synchronization transition in optomechanical systems,” Phys. Rev. A 90(5), 053810 (2014).
[Crossref]

M. Xu, D. A. Tieri, E. C. Fine, J. K. Thompson, and M. J. Holland, “Synchronization of two ensembles of atoms,” Phys. Rev. Lett. 113, 154101 (2014).
[Crossref] [PubMed]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391 (2014).
[Crossref]

M. H. Matheny, M. Grau, L. G. Villanueva, R. B. Karabalin, M. C. Cross, and M. L. Roukes, “Phase synchronization of two anharmonic nanomechanical oscillators,” Phys. Rev. Lett. 112, 014101 (2014).
[Crossref] [PubMed]

2013 (6)

V. Vlasov and A. Pikovsky, “Synchronization of a Josephson junction array in terms of global variables,” Phys. Rev. E 88, 022908 (2013).
[Crossref]

D. K. Agrawal, J. Woodhouse, and A. A. Seshia, “Observation of locked phase dynamics and enhanced frequency stability in synchronized micromechanical oscillators,” Phys. Rev. Lett. 111, 084101 (2013).
[Crossref] [PubMed]

Q. Chen, Y.-C. Lai, J. Chae, and Y. Do, “Anti-phase synchronization in microelectromechanical systems and effect of impulsive perturbations,” Phys. Rev. B 87, 144304 (2013).
[Crossref]

D. Hrg, “Synchronization of two Hindmarsh–Rose neurons with unidirectional coupling,” Neural Netw. 40, 73–79 (2013).
[Crossref] [PubMed]

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic cavity synchronization of nanomechanical oscillators,” Phys. Rev. Lett. 111(21), 213902 (2013).
[Crossref] [PubMed]

H. Tan, L. F. Buchmann, H. Seok, and G. Li, “Achieving steady-state entanglement of remote micromechanical oscillators by cascaded cavity coupling,” Phys. Rev. A 87, 022318 (2013).
[Crossref]

2012 (4)

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]

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109(23), 233906 (2012).
[Crossref]

M. C. Cross, “Improving the frequency precision of oscillators by synchronization,” Phys. Rev. E 85, 046214 (2012).
[Crossref]

I. Bargatin, E. Myers, J. Aldridge, C. Marcoux, P. Brianceau, L. Duraffourg, E. Colinet, S. Hentz, P. Andreucci, and M. Roukes, “Large-scale integration of nanoelectromechanical systems for gas sensing applications,” Nano Lett. 12(3), 1269–1274 (2012).
[Crossref] [PubMed]

2011 (3)

G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, “Collective dynamics in optomechanical arrays,” Phys. Rev. Lett. 107(4), 043603 (2011).
[Crossref] [PubMed]

K. O. Menzel, O. Arp, and A. Piel, “Chain of coupled Van der Pol oscillators as model system for density waves in dusty plasmas,” Phys. Rev. E 84, 016405 (2011).
[Crossref]

C.-L. Zou, X.-B. Zou, F.-W. Sun, Z.-F. Han, and G.-C. Guo, “Room-temperature steady-state optomechanical entanglement on a chip,” Phys. Rev. A 84, 032317 (2011).
[Crossref]

2010 (2)

S. Knnz, M. Herrmann, V. Batteiger, G. Saathoff, T. W. Hansch, K. Vahala, and T. Udem, “Injection locking of a trapped-ion phonon laser,” Phys. Rev. Lett. 105, 013004 (2010).
[Crossref]

Y.-Q. Che, J. Wang, K.-M. Tsang, and W.-L. Chan, “Unidirectional synchronization for Hindmarsh–Rose neurons via robust adaptive sliding mode control,” Nonlinear Anal.-Real World Appl. 11(2), 1096–1104 (2010).
[Crossref]

2009 (1)

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. 103, 223901 (2009).
[Crossref]

2008 (2)

M. Hossein-Zadeh and K. J. Vahala, “Observation of injection locking in an optomechanical rf oscillator,” Appl. Phys. Lett. 93(19), 191115 (2008).
[Crossref]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321(5893), 1172–1176 (2008).
[Crossref] [PubMed]

2007 (1)

2006 (2)

F. Marquardt, J. G. E. Harris, and S. M. Girvin, “Dynamical multistability induced by radiation pressure in high-finesse micromechanical optical cavities,” Phys. Rev. Lett. 96, 103901 (2006).
[Crossref] [PubMed]

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

2005 (1)

O. Cornejo-Pérez and R. Femat, “Unidirectional synchronization of Hodgkin–Huxley neurons,” Chaos Solitons Fractals 25(1), 43–53 (2005).
[Crossref]

2004 (2)

F. Sivrikaya and B. Yener, “Time synchronization in sensor networks: a survey,” IEEE Netw. 18(4), 45–50 (2004).
[Crossref]

B. Razavi, “A study of injection locking and pulling in oscillators,” IEEE J. Solid-State Circuit 39(9), 1415–1424 (2004).
[Crossref]

2003 (1)

D. Armani, T. Kippenberg, S. Spillane, and K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature (London) 421(6926), 925–928 (2003).
[Crossref]

1999 (1)

A. B. Cawthorne, P. Barbara, S. V. Shitov, C. J. Lobb, K. Wiesenfeld, and A. Zangwill, “Synchronized oscillations in Josephson junction arrays: the role of distributed coupling,” Phys. Rev. B 60, 7575–7578 (1999).
[Crossref]

1997 (1)

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997).
[Crossref]

1995 (1)

T. Klinger, F. Greiner, A. Rohde, A. Piel, and M. E. Koepke, “Van der Pol behavior of relaxation oscillations in a periodically driven thermionic discharge,” Phys. Rev. E 52, 4316–4327 (1995).
[Crossref]

1993 (1)

C. W. Gardiner, “Driving a quantum system with the output field from another driven quantum system,” Phys. Rev. Lett. 70, 2269–2272 (1993).
[Crossref] [PubMed]

1991 (1)

M. E. Koepke and D. M. Hartley, “Experimental verification of periodic pulling in a nonlinear electronic oscillator,” Phys. Rev. A 44, 6877–6887 (1991).
[Crossref] [PubMed]

1968 (1)

J. Buck and E. Buck, “Mechanism of rhythmic synchronous flashing of fireflies: fireflies of Southeast Asia may use anticipatory time-measuring in synchronizing their flashing,” Science 159, 1319–1327 (1968).
[Crossref] [PubMed]

Adler, R.

R. Adler, “A study of locking phenomena in oscillators,” in Proceedings of the I.R.E. and Waves and Electrons (1946), pp.351–357.

Agrawal, D. K.

D. K. Agrawal, J. Woodhouse, and A. A. Seshia, “Observation of locked phase dynamics and enhanced frequency stability in synchronized micromechanical oscillators,” Phys. Rev. Lett. 111, 084101 (2013).
[Crossref] [PubMed]

Aldridge, J.

I. Bargatin, E. Myers, J. Aldridge, C. Marcoux, P. Brianceau, L. Duraffourg, E. Colinet, S. Hentz, P. Andreucci, and M. Roukes, “Large-scale integration of nanoelectromechanical systems for gas sensing applications,” Nano Lett. 12(3), 1269–1274 (2012).
[Crossref] [PubMed]

Andreucci, P.

I. Bargatin, E. Myers, J. Aldridge, C. Marcoux, P. Brianceau, L. Duraffourg, E. Colinet, S. Hentz, P. Andreucci, and M. Roukes, “Large-scale integration of nanoelectromechanical systems for gas sensing applications,” Nano Lett. 12(3), 1269–1274 (2012).
[Crossref] [PubMed]

Antonio, D.

D. Antonio, D. A. Czaplewski, J. R. Guest, D. Lpez, S. I. Arroyo, and D. H. Zanette, “Nonlinearity–induced synchronization enhancement in micromechanical oscillators,” Phys. Rev. Lett. 114, 034103 (2015).
[Crossref]

Armani, D.

D. Armani, T. Kippenberg, S. Spillane, and K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature (London) 421(6926), 925–928 (2003).
[Crossref]

Arp, O.

K. O. Menzel, O. Arp, and A. Piel, “Chain of coupled Van der Pol oscillators as model system for density waves in dusty plasmas,” Phys. Rev. E 84, 016405 (2011).
[Crossref]

Arroyo, S. I.

D. Antonio, D. A. Czaplewski, J. R. Guest, D. Lpez, S. I. Arroyo, and D. H. Zanette, “Nonlinearity–induced synchronization enhancement in micromechanical oscillators,” Phys. Rev. Lett. 114, 034103 (2015).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity Optomechanics: Nano-and Micromechanical Resonators Interacting with Light (Springer-Verlag, 2014).

Bagheri, M.

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic cavity synchronization of nanomechanical oscillators,” Phys. Rev. Lett. 111(21), 213902 (2013).
[Crossref] [PubMed]

Bahl, G.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

Barbara, P.

A. B. Cawthorne, P. Barbara, S. V. Shitov, C. J. Lobb, K. Wiesenfeld, and A. Zangwill, “Synchronized oscillations in Josephson junction arrays: the role of distributed coupling,” Phys. Rev. B 60, 7575–7578 (1999).
[Crossref]

Bargatin, I.

I. Bargatin, E. Myers, J. Aldridge, C. Marcoux, P. Brianceau, L. Duraffourg, E. Colinet, S. Hentz, P. Andreucci, and M. Roukes, “Large-scale integration of nanoelectromechanical systems for gas sensing applications,” Nano Lett. 12(3), 1269–1274 (2012).
[Crossref] [PubMed]

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(23), 233906 (2012).
[Crossref]

Baskin, I.

K. Shlomi, D. Yuvaraj, I. Baskin, O. Suchoi, R. Winik, and E. Buks, “Synchronization in an optomechanical cavity,” Phys. Rev. E 91(3), 032910 (2015).
[Crossref]

Batteiger, V.

S. Knnz, M. Herrmann, V. Batteiger, G. Saathoff, T. W. Hansch, K. Vahala, and T. Udem, “Injection locking of a trapped-ion phonon laser,” Phys. Rev. Lett. 105, 013004 (2010).
[Crossref]

Bregni, S.

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

Brianceau, P.

I. Bargatin, E. Myers, J. Aldridge, C. Marcoux, P. Brianceau, L. Duraffourg, E. Colinet, S. Hentz, P. Andreucci, and M. Roukes, “Large-scale integration of nanoelectromechanical systems for gas sensing applications,” Nano Lett. 12(3), 1269–1274 (2012).
[Crossref] [PubMed]

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S. Knnz, M. Herrmann, V. Batteiger, G. Saathoff, T. W. Hansch, K. Vahala, and T. Udem, “Injection locking of a trapped-ion phonon laser,” Phys. Rev. Lett. 105, 013004 (2010).
[Crossref]

Schliesser, A.

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

Seok, H.

H. Tan, L. F. Buchmann, H. Seok, and G. Li, “Achieving steady-state entanglement of remote micromechanical oscillators by cascaded cavity coupling,” Phys. Rev. A 87, 022318 (2013).
[Crossref]

Seshia, A. A.

D. K. Agrawal, J. Woodhouse, and A. A. Seshia, “Observation of locked phase dynamics and enhanced frequency stability in synchronized micromechanical oscillators,” Phys. Rev. Lett. 111, 084101 (2013).
[Crossref] [PubMed]

Shah, S.

M. Zhang, S. Shah, J. Cardenas, and M. Lipson, “Synchronization and phase noise reduction in micromechanical oscillator arrays coupled through light,” Phys. Rev. Lett. 115, 163902 (2015).
[Crossref] [PubMed]

Shah, S. Y.

S. Y. Shah, M. Zhang, R. Rand, and M. Lipson, “Master-slave locking of optomechanical oscillators over a long distance,” Phys. Rev. Lett. 114(11), 113602 (2015).
[Crossref] [PubMed]

Shen, Z.

C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Shitov, S. V.

A. B. Cawthorne, P. Barbara, S. V. Shitov, C. J. Lobb, K. Wiesenfeld, and A. Zangwill, “Synchronized oscillations in Josephson junction arrays: the role of distributed coupling,” Phys. Rev. B 60, 7575–7578 (1999).
[Crossref]

Shkarin, A.

D. Lee, M. Underwood, D. Mason, A. Shkarin, S. Hoch, and J. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Commun. 6, 6232 (2015).
[Crossref] [PubMed]

Shlomi, K.

K. Shlomi, D. Yuvaraj, I. Baskin, O. Suchoi, R. Winik, and E. Buks, “Synchronization in an optomechanical cavity,” Phys. Rev. E 91(3), 032910 (2015).
[Crossref]

Sivrikaya, F.

F. Sivrikaya and B. Yener, “Time synchronization in sensor networks: a survey,” IEEE Netw. 18(4), 45–50 (2004).
[Crossref]

Song, H.

W. Li, C. Li, and H. Song, “Criterion of quantum synchronization and controllable quantum synchronization based on an optomechanical system,” J. Phys. B 48(3), 035503 (2015).
[Crossref]

Spillane, S.

D. Armani, T. Kippenberg, S. Spillane, and K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature (London) 421(6926), 925–928 (2003).
[Crossref]

Strogatz, S.

S. Strogatz, Sync: The Emerging Science of Spontaneous Order (Hyperion, 2003).

Suchoi, O.

K. Shlomi, D. Yuvaraj, I. Baskin, O. Suchoi, R. Winik, and E. Buks, “Synchronization in an optomechanical cavity,” Phys. Rev. E 91(3), 032910 (2015).
[Crossref]

Sun, F.-W.

C.-L. Zou, X.-B. Zou, F.-W. Sun, Z.-F. Han, and G.-C. Guo, “Room-temperature steady-state optomechanical entanglement on a chip,” Phys. Rev. A 84, 032317 (2011).
[Crossref]

Tan, H.

H. Tan, L. F. Buchmann, H. Seok, and G. Li, “Achieving steady-state entanglement of remote micromechanical oscillators by cascaded cavity coupling,” Phys. Rev. A 87, 022318 (2013).
[Crossref]

Tang, H. X.

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic cavity synchronization of nanomechanical oscillators,” Phys. Rev. Lett. 111(21), 213902 (2013).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. 103, 223901 (2009).
[Crossref]

Thompson, J. K.

M. Xu, D. A. Tieri, E. C. Fine, J. K. Thompson, and M. J. Holland, “Synchronization of two ensembles of atoms,” Phys. Rev. Lett. 113, 154101 (2014).
[Crossref] [PubMed]

Tieri, D. A.

M. Xu, D. A. Tieri, E. C. Fine, J. K. Thompson, and M. J. Holland, “Synchronization of two ensembles of atoms,” Phys. Rev. Lett. 113, 154101 (2014).
[Crossref] [PubMed]

Tsang, K.-M.

Y.-Q. Che, J. Wang, K.-M. Tsang, and W.-L. Chan, “Unidirectional synchronization for Hindmarsh–Rose neurons via robust adaptive sliding mode control,” Nonlinear Anal.-Real World Appl. 11(2), 1096–1104 (2010).
[Crossref]

Udem, T.

S. Knnz, M. Herrmann, V. Batteiger, G. Saathoff, T. W. Hansch, K. Vahala, and T. Udem, “Injection locking of a trapped-ion phonon laser,” Phys. Rev. Lett. 105, 013004 (2010).
[Crossref]

Underwood, M.

D. Lee, M. Underwood, D. Mason, A. Shkarin, S. Hoch, and J. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Commun. 6, 6232 (2015).
[Crossref] [PubMed]

Vahala, K.

S. Knnz, M. Herrmann, V. Batteiger, G. Saathoff, T. W. Hansch, K. Vahala, and T. Udem, “Injection locking of a trapped-ion phonon laser,” Phys. Rev. Lett. 105, 013004 (2010).
[Crossref]

D. Armani, T. Kippenberg, S. Spillane, and K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature (London) 421(6926), 925–928 (2003).
[Crossref]

Vahala, K. J.

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321(5893), 1172–1176 (2008).
[Crossref] [PubMed]

M. Hossein-Zadeh and K. J. Vahala, “Observation of injection locking in an optomechanical rf oscillator,” Appl. Phys. Lett. 93(19), 191115 (2008).
[Crossref]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007).
[Crossref] [PubMed]

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

Villanueva, L. G.

M. H. Matheny, M. Grau, L. G. Villanueva, R. B. Karabalin, M. C. Cross, and M. L. Roukes, “Phase synchronization of two anharmonic nanomechanical oscillators,” Phys. Rev. Lett. 112, 014101 (2014).
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V. Vlasov and A. Pikovsky, “Synchronization of a Josephson junction array in terms of global variables,” Phys. Rev. E 88, 022908 (2013).
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D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 2007).

Wang, H.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

Wang, J.

Y.-Q. Che, J. Wang, K.-M. Tsang, and W.-L. Chan, “Unidirectional synchronization for Hindmarsh–Rose neurons via robust adaptive sliding mode control,” Nonlinear Anal.-Real World Appl. 11(2), 1096–1104 (2010).
[Crossref]

Weiderhecker, G.

S. Manipatruni, G. Weiderhecker, and M. Lipson, “Long-range synchronization of optomechanical structures,” in CLEO:2011 - Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper QWI1.
[Crossref]

Wiederhecker, G. S.

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109(23), 233906 (2012).
[Crossref]

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A. B. Cawthorne, P. Barbara, S. V. Shitov, C. J. Lobb, K. Wiesenfeld, and A. Zangwill, “Synchronized oscillations in Josephson junction arrays: the role of distributed coupling,” Phys. Rev. B 60, 7575–7578 (1999).
[Crossref]

Winik, R.

K. Shlomi, D. Yuvaraj, I. Baskin, O. Suchoi, R. Winik, and E. Buks, “Synchronization in an optomechanical cavity,” Phys. Rev. E 91(3), 032910 (2015).
[Crossref]

Woodhouse, J.

D. K. Agrawal, J. Woodhouse, and A. A. Seshia, “Observation of locked phase dynamics and enhanced frequency stability in synchronized micromechanical oscillators,” Phys. Rev. Lett. 111, 084101 (2013).
[Crossref] [PubMed]

Xu, M.

M. Xu, D. A. Tieri, E. C. Fine, J. K. Thompson, and M. J. Holland, “Synchronization of two ensembles of atoms,” Phys. Rev. Lett. 113, 154101 (2014).
[Crossref] [PubMed]

Yener, B.

F. Sivrikaya and B. Yener, “Time synchronization in sensor networks: a survey,” IEEE Netw. 18(4), 45–50 (2004).
[Crossref]

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L. Ying, Y.-C. Lai, and C. Grebogi, “Quantum manifestation of a synchronization transition in optomechanical systems,” Phys. Rev. A 90(5), 053810 (2014).
[Crossref]

Yuvaraj, D.

K. Shlomi, D. Yuvaraj, I. Baskin, O. Suchoi, R. Winik, and E. Buks, “Synchronization in an optomechanical cavity,” Phys. Rev. E 91(3), 032910 (2015).
[Crossref]

Zanette, D. H.

D. Antonio, D. A. Czaplewski, J. R. Guest, D. Lpez, S. I. Arroyo, and D. H. Zanette, “Nonlinearity–induced synchronization enhancement in micromechanical oscillators,” Phys. Rev. Lett. 114, 034103 (2015).
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A. B. Cawthorne, P. Barbara, S. V. Shitov, C. J. Lobb, K. Wiesenfeld, and A. Zangwill, “Synchronized oscillations in Josephson junction arrays: the role of distributed coupling,” Phys. Rev. B 60, 7575–7578 (1999).
[Crossref]

Zhang, M.

S. Y. Shah, M. Zhang, R. Rand, and M. Lipson, “Master-slave locking of optomechanical oscillators over a long distance,” Phys. Rev. Lett. 114(11), 113602 (2015).
[Crossref] [PubMed]

M. Zhang, S. Shah, J. Cardenas, and M. Lipson, “Synchronization and phase noise reduction in micromechanical oscillator arrays coupled through light,” Phys. Rev. Lett. 115, 163902 (2015).
[Crossref] [PubMed]

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109(23), 233906 (2012).
[Crossref]

Zhang, Y.-L.

C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
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J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997).
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C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
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C.-L. Zou, X.-B. Zou, F.-W. Sun, Z.-F. Han, and G.-C. Guo, “Room-temperature steady-state optomechanical entanglement on a chip,” Phys. Rev. A 84, 032317 (2011).
[Crossref]

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C.-L. Zou, X.-B. Zou, F.-W. Sun, Z.-F. Han, and G.-C. Guo, “Room-temperature steady-state optomechanical entanglement on a chip,” Phys. Rev. A 84, 032317 (2011).
[Crossref]

Appl. Phys. Lett. (1)

M. Hossein-Zadeh and K. J. Vahala, “Observation of injection locking in an optomechanical rf oscillator,” Appl. Phys. Lett. 93(19), 191115 (2008).
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W. Li, C. Li, and H. Song, “Criterion of quantum synchronization and controllable quantum synchronization based on an optomechanical system,” J. Phys. B 48(3), 035503 (2015).
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D. Lee, M. Underwood, D. Mason, A. Shkarin, S. Hoch, and J. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Commun. 6, 6232 (2015).
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C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
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Nat. Phys. (1)

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

Nature (London) (1)

D. Armani, T. Kippenberg, S. Spillane, and K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature (London) 421(6926), 925–928 (2003).
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D. Hrg, “Synchronization of two Hindmarsh–Rose neurons with unidirectional coupling,” Neural Netw. 40, 73–79 (2013).
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Y.-Q. Che, J. Wang, K.-M. Tsang, and W.-L. Chan, “Unidirectional synchronization for Hindmarsh–Rose neurons via robust adaptive sliding mode control,” Nonlinear Anal.-Real World Appl. 11(2), 1096–1104 (2010).
[Crossref]

Opt. Express (1)

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L. Ying, Y.-C. Lai, and C. Grebogi, “Quantum manifestation of a synchronization transition in optomechanical systems,” Phys. Rev. A 90(5), 053810 (2014).
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C.-L. Zou, X.-B. Zou, F.-W. Sun, Z.-F. Han, and G.-C. Guo, “Room-temperature steady-state optomechanical entanglement on a chip,” Phys. Rev. A 84, 032317 (2011).
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Phys. Rev. Lett. (15)

F. Marquardt, J. G. E. Harris, and S. M. Girvin, “Dynamical multistability induced by radiation pressure in high-finesse micromechanical optical cavities,” Phys. Rev. Lett. 96, 103901 (2006).
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M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. 103, 223901 (2009).
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S. Knnz, M. Herrmann, V. Batteiger, G. Saathoff, T. W. Hansch, K. Vahala, and T. Udem, “Injection locking of a trapped-ion phonon laser,” Phys. Rev. Lett. 105, 013004 (2010).
[Crossref]

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109(23), 233906 (2012).
[Crossref]

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic cavity synchronization of nanomechanical oscillators,” Phys. Rev. Lett. 111(21), 213902 (2013).
[Crossref] [PubMed]

M. Zhang, S. Shah, J. Cardenas, and M. Lipson, “Synchronization and phase noise reduction in micromechanical oscillator arrays coupled through light,” Phys. Rev. Lett. 115, 163902 (2015).
[Crossref] [PubMed]

S. Y. Shah, M. Zhang, R. Rand, and M. Lipson, “Master-slave locking of optomechanical oscillators over a long distance,” Phys. Rev. Lett. 114(11), 113602 (2015).
[Crossref] [PubMed]

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
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Figures (8)

Fig. 1
Fig. 1 Schematic setup of the unidirectionally cascaded systems of two toroid optical microcavities, coupled through a unidirectional fiber with light. The distance between the two OMSs is denoted as L.
Fig. 2
Fig. 2 Numerical solutions of equations of motion in the unidirectionally cascaded two OMSs scheme, for the parameters: ωm1 = 1, ωm2 = 1.005, Δ1 = −ωm1, Δ2 = −ωm2, G1 = G2 = 4 × 10, κ1 = κ2 = 0.15, κex1 = κex2 = 0 075, γm1 = γm2 = 5 × 10−3. (a) and (e) Dynamical evolution. (b) and (f) Power spectrum density (PSD) of the displacement operators q1 and q2. (c) and (g) Phase diagram of q1, q2. (d) and (h) Displacement PSD of each single OMS driven by a constant amplitude optical field E. The left and right columns correspond to E = 64 and E = 100.
Fig. 3
Fig. 3 (a) The PSD of q1 as a function of the effective driving strength E, with a sole frequency peak denoted as ω m 1 . (b) The PSD of q2 relative to ω m 1 as a function of the effective driving strength E. The maximum frequency component in the PSD of q2 without the frequency shift is denoted as ω m 2 . The color scaled in the color bar indicates the power values in the PSD. The other simulation parameters are the same as those in Fig. 2.
Fig. 4
Fig. 4 The final relative frequency difference between maximum frequency components of two OMSs vs the relative intrinsic frequency mismatch (ωm2ωm1)m1. (a) κex1 = 0.075 (κ1 = 0.15), blue line: E = 40, red line: E = 64, black real line: uncoupled free-running case. (b) E = 64, blue line: κex1 = 0.045 (κ1 = 0.12), red line: κex1 = 0.09 (κ1 = 0.165), green line: κex1 = 0.135 (κ1 = 0.21). The gray domain represents the synchronization region. The other simulation parameters are the same as those in Fig. 2.
Fig. 5
Fig. 5 Considering fiber-loss, (a) the final relative frequency difference between maximum frequency components of the two OMSs ( ω m 2 ω m 1 ) / ω m 1 vs relative intrinsic frequency mismatch the (ωm2ωm1)m1 with L = 4.−6km, η12 = 0.9. The cases corresponding to the lines are the same as those in Fig. 4(a). (b) The final maximum frequency components of the two OMSs ω m 1 , ω m 2 vs distance L. Blue line marked with x : ω m 2 ( E = 40 ) . Red line marked with o : ω m 2 ( E = 64 ) . Blue dash-dotted line: ω m 1 ( E = 40 ) . Red dashed line: ω m 1 ( E = 64 ) .
Fig. 6
Fig. 6 Schematic setup of the unidirectionally cascaded synchronization scheme consisting of three OMSs. The distance between the first (last) two OMSs is denoted as L1 (L2).
Fig. 7
Fig. 7 Numerical solutions of the equations of motion in the unidirectionally cascaded three OMSs scheme. (a) and (b) The PSDs of q1, q2 and q3. The parameters are: (a) ωm2 = 0.995, ωm3 = 1.005, (b) ωm2 = 1.005, ωm3 = 1.010. (c) and (d) The synchronization region of ωm3 relative to ωm1. The parameters are: (c) ωm1 = 1, ωm2 = 0.995, (d) ωm1 = 1, ωm2 = 1.005.
Fig. 8
Fig. 8 The PSDs of the output optical field of OMS-2: (a) E = 30, (b) E = 64, (c) E = 100. The other simulation parameters are the same as those in Fig. 2.

Equations (12)

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H j = h ¯ ω c j a j a j + h ¯ ω m j b j b j h ¯ g j a j a j ( b j + b j ) ,
a ˙ j = 1 i h ¯ [ a j , H j ] 1 2 κ j a j + κ e x j 1 / 2 a i n ( j ) , b ˙ j = 1 i h ¯ [ b j , H j ] 1 2 γ m j b j ,
a i n ( 2 ) ( t ) = η 12 a o u t ( 1 ) ( t τ ) ,
a o u t ( 1 ) ( t ) = a i n ( 1 ) ( t ) κ e x 1 1 / 2 a 1 .
a ˜ ˙ 1 = ( i Δ 1 + 1 2 κ 1 ) a ˜ 1 + i g 1 a ˜ 1 ( b 1 + b 1 ) + E ,
a ˜ ˙ 2 = ( i Δ 2 + 1 2 κ 2 ) a ˜ 2 + i g 2 a ˜ 2 ( b 2 + b 2 ) + η 12 κ e x 2 1 / 2 ( κ e x 1 1 / 2 E κ e x 1 1 / 2 a ˜ 1 ) ,
b ˙ J = i ω m j b j + i g j a ˜ j a ˜ j 1 2 γ m j b j , j = 1 , 2 ,
ρ ˙ = 1 i h ¯ [ H 1 + H 2 , ρ ] + κ 1 [ a 1 ] ρ + κ 2 [ a 2 ] ρ + γ m 1 [ b 1 ] ρ + γ m 2 [ b 2 ] ρ + κ e x 1 κ e x 2 × η 12 { [ a 1 ρ , a 2 ] + [ a 2 , ρ a 1 ] } + [ a i n ( 1 ) ( κ e x 1 a 1 + η 12 κ e x 2 a 2 ) h . c , ρ ] ,
Q j = ( a ˜ j + a ˜ j ) / 2 , P j = i ( a ˜ j a ˜ j ) / 2 , q j = ( b j + b j ) / 2 1 / 2 , P j = i ( b j b j ) / 2 1 / 2 ,
Q ˙ 1 = ( Δ 1 G 1 q 1 ) P 1 1 2 κ 1 Q 1 + E , P ˙ 1 = ( Δ 1 G 1 q 1 ) Q 1 1 2 κ 1 P 1 , Q ˙ 2 = ( Δ 2 G 2 q 2 ) P 2 1 2 κ 2 Q 2 η 12 κ e x 2 1 / 2 ( κ e x 1 1 / 2 E κ e x 1 1 / 2 Q 1 ) , P ˙ 2 = ( Δ 2 G 2 q 2 ) Q 2 1 2 κ 2 P 2 η 12 ( κ e x 2 κ e x 1 ) 1 / 2 P 1 , q ˙ j = ω m j p j , p ˙ j = ω m j q j γ m j p j + G j ( Q j 2 + P j 2 ) ,
a ˜ ˙ 3 = ( i Δ 3 + 1 2 κ 3 ) a ˜ 3 + i g 3 a ˜ 3 ( b 3 + b 3 ) + η 23 η 12 κ e x 3 1 / 2 ( κ e x 1 1 / 2 E κ e x 1 1 / 2 a ˜ 1 ) η 23 ( κ e x 3 κ e x 2 ) 1 / 2 a ˜ 2 ,
b ˙ 3 = i ω m 3 b 3 + i g 3 a ˜ 3 a ˜ 3 1 2 γ m 3 b 3 .

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