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Quantum Correlations and Synchronization Measures

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Lectures on General Quantum Correlations and their Applications

Part of the book series: Quantum Science and Technology ((QST))

Abstract

The phenomenon of spontaneous synchronization is universal and only recently advances have been made in the quantum domain. Being synchronization a kind of temporal correlation among systems, it is interesting to understand its connection with other measures of quantum correlations. We review here what is known in the field, putting emphasis on measures and indicators of synchronization which have been proposed in the literature, and comparing their validity for different dynamical systems, highlighting when they give similar insights and when they seem to fail.

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References

  1. A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge edition, 2001)

    Google Scholar 

  2. S.H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Westview p edition, 2001)

    Google Scholar 

  3. R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. K. Modi, A. Brodutch, H. Cable, T. Paterek, V. Vedral, The classical-quantum boundary for correlations: discord and related measures. Rev. Mod. Phys. 84, 1655–1707 (2012)

    Article  ADS  Google Scholar 

  5. G. Adesso, T.R. Bromley, M. Cianciaruso, Measures and applications of quantum correlations. J. Phys. A: Math. Theor. 49, 473001 (2016)

    Google Scholar 

  6. S. Boccaletti, J. Kurths, G. Osipov, D.L. Valladares, C.S. Zhou, The synchronization of chaotic systems. Phys. Rep. 366, 1–101 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. I. Goychuk, J. Casado-Pascual, M. Morillo, J. Lehmann, P. Hänggi, Quantum stochastic synchronization. Phys. Rev. Lett. 97(21), 210601 (2006)

    Article  ADS  Google Scholar 

  8. O. Zhirov, D. Shepelyansky, Synchronization and bistability of a qubit coupled to a driven dissipative oscillator. Phys. Rev. Lett. 100(1), 014101 (2008)

    Article  ADS  Google Scholar 

  9. O. Zhirov, D. Shepelyansky, Quantum synchronization and entanglement of two qubits coupled to a driven dissipative resonator. Phys. Rev. B 80(1), 014519 (2009)

    Article  ADS  Google Scholar 

  10. T.E. Lee, H.R. Sadeghpour, Quantum synchronization of quantum van der Pol oscillators with trapped Ions. Phys. Rev. Lett. 111, 234101 (2013)

    Google Scholar 

  11. S. Walter, A. Nunnenkamp, C. Bruder, Quantum synchronization of a driven self-sustained oscillator. Phys. Rev. Lett. 112(9), 094102 (2014)

    Article  ADS  Google Scholar 

  12. P.P. Orth, D. Roosen, W. Hofstetter, K. Le Hur, Dynamics, synchronization, and quantum phase transitions of two dissipative spins. Phys. Rev. B 82, 144423 (2010)

    Google Scholar 

  13. G. Heinrich, M. Ludwig, J. Qian, B. Kubala, F. Marquardt, Collective dynamics in optomechanical arrays. Phys. Rev. Lett. 107(4), 043603 (2011)

    Article  ADS  Google Scholar 

  14. C.A. Holmes, C.P. Meaney, G.J. Milburn, Synchronization of many nanomechanical resonators coupled via a common cavity field. Phys. Rev. E 85, 066203 (2012)

    Google Scholar 

  15. G.L. Giorgi, F. Galve, G. Manzano, P. Colet, R. Zambrini, Quantum correlations and mutual synchronization. Phys. Rev. A 85, 052101 (2012)

    Google Scholar 

  16. G. Manzano, F. Galve, G.L. Giorgi, E. Hernández-García, R. Zambrini, Synchronization, quantum correlations and entanglement in oscillator networks. Sci. Rep. 3, 1439 (2013)

    Article  Google Scholar 

  17. A. Mari, A. Farace, N. Didier, V. Giovannetti, R. Fazio, Measures of quantum synchronization in continuous variable systems. Phys. Rev. Lett. 111(10), 103605 (2013)

    Article  ADS  Google Scholar 

  18. S.C. Manrubia, A.S. Mikhailov, D.H. Zanette, Emergence of Dynamical Order. Synchronization Phenomena in Complex Systems (World Scientific Publishing Co., Singapore, 2004). Lecture no edition

    Book  MATH  Google Scholar 

  19. A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Synchronization in complex networks. Phys. Rep. 469(3), 93–153 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  20. J. Pantaleone, Synchronization of metronomes. Am. J. Phys. 70(10), 992 (2002)

    Article  ADS  Google Scholar 

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

    Google Scholar 

  22. M. Aspelmeyer, T.J. Kippenberg, F. Marquardt, Cavity optomechanics. Rev. Mod. Phys. 86, 1391 (2014)

    Google Scholar 

  23. M. Ludwig, F. Marquardt, Quantum many-body dynamics in optomechanical arrays. Phys. Rev. Lett. 111(7), 073603 (2013)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  25. M. Bagheri, M. Poot, L. Fan, F. Marquardt, H.X. Tang, Photonic cavity synchronization of nanomechanical oscillators. Phys. Rev. Lett. 111, 213902 (2013)

    Google Scholar 

  26. S.Y. Shah, M. Zhang, R. Rand, M. Lipson, Master-slave locking of optomechanical oscillators over a long distance. Phys. Rev. Lett. 114, 113602 (2015)

    Google Scholar 

  27. W. Li, F. Zhang, C. Li, H. Song, Quantum synchronization in a star-type cavity QED network. Commun. Nonlinear Sci. Numer. Simul. 42, 121–131 (2017)

    Article  ADS  MathSciNet  Google Scholar 

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

    Google Scholar 

  29. M.H. Matheny, M. Grau, L.G. Villanueva, R.B. Karabalin, M.C. Cross, M.L. Roukes, Phase synchronization of two anharmonic nanomechanical oscillators. Phys. Rev. Lett. 112, 014101 (2014)

    Google Scholar 

  30. T.E. Lee, C.-K. Chan, S. Wang, Entanglement tongue and quantum synchronization of disordered oscillators. Phys. Rev. E 89(2), 022913 (2014)

    Google Scholar 

  31. S. Walter, A. Nunnenkamp, C. Bruder, Quantum synchronization of two Van der Pol oscillators. Annalen der Physik 527(1–2), 131–138 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. V. Ameri, M. Eghbali-Arani, A. Mari, A. Farace, F. Kheirandish, V. Giovannetti, R. Fazio, Mutual information as an order parameter for quantum synchronization. Phys. Rev. A 91(1), 012301 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  33. H. Carmichael, An Open Systems Approach to Quantum Optics: Lectures Presented at the Université Libre de Bruxelles, October 28 to November 4, 1991, Lecture Notes in Physics Monographs (Springer, Berlin, 2009)

    Google Scholar 

  34. G. Manzano, F. Galve, R. Zambrini, Avoiding dissipation in a system of three quantum harmonic oscillators. Phys. Rev. A 87(3), 032114 (2013)

    Google Scholar 

  35. G.M. Xue, M. Gong, H.K. Xu, W.Y. Liu, H. Deng, Y. Tian, H.F. Yu, Y. Yu, D.N. Zheng, S.P. Zhao, S. Han, Observation of quantum stochastic synchronization in a dissipative quantum system. Phys. Rev. B 90, 224505 (2014)

    Article  ADS  Google Scholar 

  36. G.L. Giorgi, F. Plastina, G. Francica, R. Zambrini, Spontaneous synchronization and quantum correlation dynamics of open spin systems. Phys. Rev. A 88(4), 042115 (2013)

    Google Scholar 

  37. D. Viennot, L. Aubourg, Quantum chimera states. Phys. Lett. A. 380(5–6), 678–683 (2016)

    Google Scholar 

  38. X. Minghui, D.A. Tieri, E.C. Fine, J.K. Thompson, M.J. Holland, Synchronization of two ensembles of atoms. Phys. Rev. Lett. 113, 154101 (2014)

    Article  ADS  Google Scholar 

  39. B. Zhu, J. Schachenmayer, M. Xu, F. Herrera, J.G. Restrepo, M.J. Holland, A.M. Rey, Synchronization of interacting quantum dipoles. New J. Phys. 17, 083063 (2015)

    Google Scholar 

  40. C. Deutsch, F. Ramirez-Martinez, C. Lacroûte, F. Reinhard, T. Schneider, J.N. Fuchs, F. Piéchon, F. Laloë, J. Reichel, P. Rosenbusch, Spin self-rephasing and very long coherence times in a trapped atomic ensemble. Phys. Rev. Lett. 105(2), 020401 (2010)

    Article  ADS  Google Scholar 

  41. Y. Liu, F. Piéchon, J.N. Fuchs, Quantum loss of synchronization in the dynamics of two spins. EPL (Europhys. Lett.) 103(1), 17007 (2013)

    Article  ADS  Google Scholar 

  42. M.R. Hush, W. Li, S. Genway, I. Lesanovsky, A.D. Armour, Spin correlations as a probe of quantum synchronization in trapped-ion phonon lasers. Phys. Rev. A 91, 061401(R) (2015)

    Google Scholar 

  43. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C.R. Mirasso, L. Pesquera, K. Alan Shore, Chaos-based communications at high bit rates using commercial fibre-optic links. Nature 438(7066), 343–346 (2005)

    Article  ADS  Google Scholar 

  44. G.L. Giorgi, F. Galve, R. Zambrini, Probing the spectral density of a dissipative qubit via quantum synchronization. Phys. Rev. A 94, 052121 (2016)

    Google Scholar 

  45. H. Ollivier, W.H. Zurek, Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)

    Article  ADS  MATH  Google Scholar 

  46. L. Henderson, V. Vedral, Classical, quantum and total correlations. J. Phys. A: Math. Gen. 34(35), 6899 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. V.M. Bastidas, I. Omelchenko, A. Zakharova, E. Schöll, T. Brandes, Quantum signatures of chimera states. Phys. Rev. E 92, 062924 (2015)

    Google Scholar 

  48. A.E. Motter, Nonlinear dynamics: spontaneous synchrony breaking. Nat. Phys. 6(3), 164–165 (2010)

    Article  Google Scholar 

  49. W.K. Wootters, Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998)

    Article  ADS  Google Scholar 

  50. T.E. Lee, M.C. Cross, Quantum-classical transition of correlations of two coupled cavities. Phys. Rev. A 88, 013834 (2013)

    Google Scholar 

  51. Y. Kuramoto, International Symposium on Mathematical Problems in Theoretical Physics, vol. 39 (Springer, New York, 1975)

    Book  Google Scholar 

  52. J.a. Acebrón, L.L. Bonilla, C.J. Pérez Vicente, F. Ritort, R. Spigler, The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77(1), 137–185 (2005)

    Google Scholar 

  53. I.H. de Mendoza, L.a. Pachón, J. Gómez-Gardeñes, D. Zueco, Synchronization in a semiclassical Kuramoto model. Phys. Rev. E 90(5), 052904 (2014)

    Google Scholar 

  54. K. Shlomi, D. Yuvaraj, I. Baskin, O. Suchoi, R. Winik, E. Buks, Synchronization in an optomechanical cavity. Phys. Rev. E 91, 032910 (2015)

    Google Scholar 

  55. C. Benedetti, F. Galve, A. Mandarino, M.G.A. Paris, R. Zambrini, Minimal model for spontaneous synchronization. Phys. Rev. A 94, 052118 (2016)

    Google Scholar 

  56. F. Galve, A. Mandarino, M.G.A. Paris, C. Benedetti, R. Zambrini, Microscopic description for the emergence of collective dissipation in extended quantum systems. Sci. Reps. 7, 42050 (2017)

    Google Scholar 

  57. A. Ferraro, S. Olivares, M.G.a. Paris, Gaussian states in continuous variable quantum information. (Bibliopolis, Napoli 2005; ISBN 88-7088-483-X)

    Google Scholar 

  58. G. Adesso, F. Illuminati, Entanglement in continuous-variable systems: recent advances and current perspectives. J. Phys. A 40, 7821 (2007)

    Google Scholar 

  59. E. Lieb, T. Schultz, D. Mattis, Two soluble models of an antiferromagnetic chain. Ann. Phys. 16(3), 407–466 (1961)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  60. H.P. Breuer, F. Petruccione, The Theory of Open Quantum Systems. (OUP Oxford, Oxford, 2007)

    Google Scholar 

  61. B. Bellomo, G.L. Giorgi, G.M. Palma, R. Zambrini, Quantum synchronization as a local signature of super and subradiance. arXiv:1612.07134

  62. W. Li, C. Li, H. Song, Criterion of quantum synchronization and controllable quantum synchronization based on an optomechanical system. J. Phys. B: At. Mol. Opt. Phys. 48(3), 035503 (2015)

    Article  ADS  Google Scholar 

  63. P.D. Drummond, M.D. Reid, Correlations in nondegenerate parametric oscillation. II. below threshold results. Phys. Rev. A 41(7), 3930–3949 (1990)

    Google Scholar 

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Acknowledgements

Funding from EU project QuProCS (Grant Agreement No. 641277), MINECO and FEDER/AEI (NOMAQ FIS2014-60343-P and QuStruct FIS2015-66860-P), and “Vicerectorat d’Investigació i Postgrau” of the UIB are acknowledged.

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Authors and Affiliations

  1. IFISC (UIB-CSIC), Instituto de Física Interdisciplinar y Sistemas Complejos, Palma de Mallorca, Spain

    Fernando Galve, Gian Luca Giorgi & Roberta Zambrini

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  1. Fernando Galve
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  2. Gian Luca Giorgi
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  3. Roberta Zambrini
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Correspondence to Fernando Galve or Roberta Zambrini .

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Editors and Affiliations

  1. São Paulo State University, São Paulo, Brazil

    Felipe Fernandes Fanchini

  2. São Carlos Institute of Physics, University of São Paulo, São Paulo, Brazil

    Diogo de Oliveira Soares Pinto

  3. School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom

    Gerardo Adesso

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Galve, F., Luca Giorgi, G., Zambrini, R. (2017). Quantum Correlations and Synchronization Measures. In: Fanchini, F., Soares Pinto, D., Adesso, G. (eds) Lectures on General Quantum Correlations and their Applications. Quantum Science and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-53412-1_18

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