Quantum synchronization of chaotic oscillator behaviors among coupled BEC–optomechanical systems



We consider and theoretically analyze a Bose-Einstein condensate (BEC) trapped inside an optomechanical system consisting of single-mode optical cavity with a moving end mirror. The BEC is formally analogous to a mirror driven by radiation pressure with strong nonlinear coupling. Such a nonlinear enhancement can make the oscillator display chaotic behavior. By establishing proper oscillator couplings, we find that this chaotic motion can be synchronized with other oscillators, even an oscillator network. We also discuss the scheme feasibility by analyzing recent experiment parameters. Our results provide a promising platform for the quantum signal transmission and quantum logic control, and they are of potential applications in quantum information processing and quantum networks.


Quantum synchronization Quantum chaos Cavity optomechanics Quantum networks 



All authors thank Jiong Cheng, Wenzhao Zhang and Yang Zhang for the useful discussion. This research was supported by the National Natural Science Foundation of China (Grant Nos. 11574041 and 11175033) and the Fundamental Research Funds for the Central Universities (DUT13LK05).


  1. 1.
    Vinokur, V.M., et al.: Superinsulator and quantum synchronization. Nature 452, 613–615 (2008)ADSCrossRefGoogle Scholar
  2. 2.
    Heinrich, G., Ludwig, M., Qian, J., Kubala, B., Marquardt, F.: Collective dynamics in optomechanical arrays. Phys. Rev. Lett. 107, 043603 (2011)ADSCrossRefGoogle Scholar
  3. 3.
    Zhirov, O.V., Shepelyansky, D.L.: Synchronization and bistability of a qubit coupled to a driven dissipative oscillator. Phys. Rev. Lett. 100, 014101 (2008)ADSCrossRefGoogle Scholar
  4. 4.
    Orth, P.P., Roosen, D., Hofstetter, W., Hur, K.L.: Dynamics, synchronization, and quantum phase transitions of two dissipative spins. Phys. Rev. B 82, 144423 (2010)ADSCrossRefGoogle Scholar
  5. 5.
    Zhirov, O.V., Shepelyansky, D.L.: Quantum synchronization and entanglement of two qubits coupled to a driven dissipative resonator. Phys. Rev. B 80, 014519 (2009)ADSCrossRefGoogle Scholar
  6. 6.
    Giorgi, G.L., Plastina, F., Francica, G., Zambrini, R.: Spontaneous synchronization and quantum correlation dynamics of open spin systems. Phys. Rev. A 88, 042115 (2013)ADSCrossRefGoogle Scholar
  7. 7.
    Lee, T.E., Sadeghpour, H.R.: Quantum synchronization of quantum van der Pol oscillators with trapped ions. Phys. Rev. Lett. 111, 234101 (2013)ADSCrossRefGoogle Scholar
  8. 8.
    Xu, M.H., Tieri, D.A., Fine, E.C., Thompson, J.K., Holland, M.J.: Synchronization of two ensembles of atoms. Phys. Rev. Lett. 113, 154101 (2014)ADSCrossRefGoogle Scholar
  9. 9.
    Walter, S., Nunnenkamp, A., Bruder, C.: Quantum synchronization of a driven self-sustained oscillator. Phys. Rev. Lett. 112, 094102 (2014)ADSCrossRefGoogle Scholar
  10. 10.
    Ying, L., Lai, Y.C., Grebogi, C.: Quantum manifestation of a synchronization transition in optomechanical systems. Phys. Rev. A 90, 053810 (2014)ADSCrossRefGoogle Scholar
  11. 11.
    Shlomi, K., et al.: Synchronization in an optomechanical cavity. Phys. Rev. E 91, 032910 (2015)ADSCrossRefGoogle Scholar
  12. 12.
    Samoylova, M., Piovella, N.M., Robb, G.R., Bachelard, R., Courteille, Ph.W.: Synchronization of Bloch oscillations by a ring cavity. arXiv:1503.05616
  13. 13.
    Mari, A., Farace, A., Didier, N., Giovannetti, V., Fazio, R.: Measures of quantum synchronization in continuous variable systems. Phys. Rev. Lett. 111, 103605 (2013)ADSCrossRefGoogle Scholar
  14. 14.
    Ameri, V., et al.: Mutual information as an order parameter for quantum synchronization. Phys. Rev. A 91, 012301 (2015)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Choi, S.H., Ha, S.Y.: Quantum synchronization of the Schrödinger-Lohe model. J. Phys. A Math. Theor. 47, 355104 (2014)CrossRefMATHGoogle Scholar
  16. 16.
    Manzano, G., Galve, F., Giorgi, G.L., Hernández-García, E., Zambrini, R.: Synchronization, quantum correlations and entanglement in oscillator networks. Sci. Rep. 3, 1439 (2013)ADSCrossRefGoogle Scholar
  17. 17.
    Li, W.L., Li, C., Song, H.S.: Quantum synchronization in an optomechanical system based on Lyapunov control. Phys. Rev. E 93, 062221 (2016)ADSCrossRefGoogle Scholar
  18. 18.
    Walter, S., Nunnenkamp, A., Bruder, C.: Quantum synchronization of two Van der Pol oscillators. Ann. Phys. (Leipzig) 527, 131 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Chacón, R., Palmero, F., Cuevas-Maraver, J.: Impulse-induced localized control of chaos in starlike networks. Phys. Rev. E 93, 062210 (2016)ADSCrossRefGoogle Scholar
  20. 20.
    Zhang, J., et al.: Quantum internet using code division multiple access. Sci. Rep. 3, 2211 (2013)ADSGoogle Scholar
  21. 21.
    Li, W.L., Li, C., Song, H.S.: Quantum parameter identification for a chaotic atom ensemble system. Phys. Lett. A 380, 672–677 (2016)ADSCrossRefMATHGoogle Scholar
  22. 22.
    Modi, K., Brodutch, A., Cable, H., Paterek, T., Vedral, V.: The classical-quantum boundary for correlations: discord and related measures. Rev. Mod. Phys. 84, 1655–1707 (2012)ADSCrossRefGoogle Scholar
  23. 23.
    Streltsov, A., Lee, S., Adesso, G.: Concentrating tripartite quantum information. Phys. Rev. Lett. 115, 030505 (2015)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Campbell, S., et al.: Global quantum correlations in finite-size spin chains. New J. Phys. 15, 043033 (2013)ADSCrossRefGoogle Scholar
  25. 25.
    Zhang, J., Zhang, Y., Yu, C.S.: entropic uncertainty relation and information exclusion relation for multiple measurements in the presence of quantum memory. Sci. Rep. 5, 11701 (2015)ADSCrossRefGoogle Scholar
  26. 26.
    Li, W.L., Zhang, F.Y., Li, C., Song, H.S.: Quantum synchronization in a star-type cavity QED network. Commun. Nonlinear Sci. Numer. Simul. 42, 121–131 (2017)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Li, W.L., Li, C., Song, H.S.: Quantum synchronization and quantum state sharing in irregular complex network. arXiv:1606.08113
  28. 28.
    Marquardt, F., Girvin, S.M.: Optomechanics. Physics 2, 40 (2009)CrossRefGoogle Scholar
  29. 29.
    Aspelmeyer, M., Kippenberg, T.J., Marquardt, F.: Cavity optomechanics. Rev. Mod. Phys. 86, 1391–1452 (2014)ADSCrossRefGoogle Scholar
  30. 30.
    Wang, G., Huang, L., Lai, Y.C., Grebogi, C.: Nonlinear dynamics and quantum entanglement in optomechanical systems. Phys. Rev. Lett. 112, 110406 (2014)ADSCrossRefGoogle Scholar
  31. 31.
    Chan, J., et al.: Laser cooling of a nanomechanical oscillator into its quantum ground state. Nature 478, 89–92 (2011)ADSCrossRefGoogle Scholar
  32. 32.
    Verhagen, E., Deléglise, S., Weis, S., Schliesser, A., Kippenberg, T.J.: Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode. Nature 482, 63–67 (2012)ADSCrossRefGoogle Scholar
  33. 33.
    Zhang, W.Z., Cheng, J., Liu, J.Y., Zhou, L.: Controlling photon transport in the single-photon weak-coupling regime of cavity optomechanics. Phys. Rev. A 91, 063836 (2015)ADSCrossRefGoogle Scholar
  34. 34.
    Zhang, Y., Zhang, J., Yu, C.S.: Photon statistics on the extreme entanglement. Sci. Rep. 6, 24098 (2016)ADSCrossRefGoogle Scholar
  35. 35.
    Cheng, J., Zhang, W.Z., Zhou, L., Zhang, W.: Preservation macroscopic entanglement of optomechanical systems in non-Markovian environment. Sci. Rep. 6, 23678 (2016)ADSCrossRefGoogle Scholar
  36. 36.
    Mari, A., Eisert, J.: Gently modulating optomechanical systems. Phys. Rev. Lett. 103, 213603 (2009)ADSCrossRefGoogle Scholar
  37. 37.
    Farace, A., Giovannetti, V.: Enhancing quantum effects via periodic modulations in optomechanical systems. Phys. Rev. A 86, 013820 (2012)ADSCrossRefGoogle Scholar
  38. 38.
    Lü, L., et al.: Determination of configuration matrix element and outer synchronization among networks with different topologies. Physica A 461, 833–839 (2016)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Lü, L., Chen, L.S., Bai, S.Y., Li, G.: A new synchronization tracking technique for uncertain discrete network with spatiotemporal chaos behaviors. Physica A 460, 314–325 (2016)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Sun, H.J., Cao, H.J.: Synchronization of two identical and non-identical Rulkov models. Commun. Nonlinear Sci. Numer. Simul. 40, 15–27 (2016)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Shepelev, I.A., Slepnev, A.V., Vadivasova, T.E.: Different synchronization characteristics of distinct types of traveling waves in a model of active medium with periodic boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 38, 206–217 (2016)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Zhang, K., Chen, W., Bhattacharya, M., Meystre, P.: Hamiltonian chaos in a coupled BEC-optomechanical-cavity system. Phys. Rev. A 81, 013802 (2010)ADSCrossRefGoogle Scholar
  43. 43.
    Yasir, K.A., Liu, W.M.: Tunable bistability in hybrid Bose-Einstein condensate optomechanics. Sci. Rep. 5, 10612 (2015)ADSCrossRefGoogle Scholar
  44. 44.
    Horak, P., Barnett, S.M., Ritsch, H.: Coherent dynamics of Bose-Einstein condensates in high-finesse optical cavities. Phys. Rev. A 61, 033609 (2000)ADSCrossRefGoogle Scholar
  45. 45.
    Vitali, D., et al.: Optomechanical entanglement between a movable mirror and a cavity field. Phys. Rev. Lett. 98, 030405 (2007)ADSCrossRefGoogle Scholar
  46. 46.
    Gardiner, C.W., Zoller, P.: Quantum Noise. Chapter 3. Springer, Berlin (2000). Please check and confirm the book title for reference [46] is correctCrossRefMATHGoogle Scholar
  47. 47.
    Giovannetti, V., Vitali, D.: Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion. Phys. Rev. A 63, 023812 (2001)ADSCrossRefGoogle Scholar
  48. 48.
    Buchmann, L.F., Wright, E.M., Meystre, P.: Phase conjugation in quantum optomechanics. Phys. Rev. A 88, 041801(R) (2013)ADSCrossRefGoogle Scholar
  49. 49.
    Weiss, T., Kronwald, A., Marquardt, F.: Noise-induced transitions in optomechanical synchronization. New. J. Phys. 18, 013043 (2016)ADSCrossRefGoogle Scholar
  50. 50.
    Bakemeier, L., Alvermann, A., Fehske, H.: Route to Chaos in optomechanics. Phys. Rev. Lett. 114, 013601 (2015)ADSCrossRefGoogle Scholar
  51. 51.
    Lü, L., Li, C., Li, G., Sun, A., Yan, Z., Rong, T., Gao, Y.: Synchronization transmission of laser pattern signal within uncertain switched network. Commun. Nonlinear. Sci. Numer. Simulat. 47, 267 (2017)ADSCrossRefGoogle Scholar
  52. 52.
    Lü, X.Y., Jing, H., Ma, J.Y., Wu, Y.: PT-symmetry-breaking chaos in optomechanics. Phys. Rev. Lett. 114, 253601 (2015)ADSCrossRefGoogle Scholar
  53. 53.
    Lü, L., Li, C.R., Chen, L.S.: Projective synchronization of the small world delayed network with uncertainty. Nonlinear Dyn. 76, 1633 (2014)ADSCrossRefGoogle Scholar
  54. 54.
    Galve, F., Giorgi, G.L., Zambrini, R.: Quantum correlations and synchronization measures. arXiv:1610.05060
  55. 55.
    Lü, L., Li, C.R., Chen, L.S., Zhao, G.N.: New technology of synchronization for the uncertain dynamical network with the switching topology. Nonlinear Dyn. 86(1), 655–666 (2016)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Schultz, P., et al.: Tweaking synchronization by connectivity modifications. Phys. Rev. E 93, 062211 (2016)ADSCrossRefGoogle Scholar
  57. 57.
    Mahata, S., Das, S., Gupte, N.: Synchronization in area-preserving maps: effects of mixed phase space and coherent structures. Phys. Rev. E 93, 062212 (2016)ADSCrossRefGoogle Scholar
  58. 58.
    Ludwig, M., Kubala, B., Marquardt, F.: The optomechanical instability in the quantum regime. New J. Phys. 10, 095013 (2008)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of Physics and Optoelectronic EngineeringDalian University of TechnologyDalianChina

Personalised recommendations