Abstract
We study an emergent synchronous behavior for an ensemble of Lohe qubit oscillators whose quantum states are described by \(2\times 2\) unitary matrices. The quantum Lohe model can be regarded as a non-abelian and quantum generalization of the Kuramoto model for classical oscillators. For the interacting qubit system, the Lohe model can be recast as a coupled ODE system. We provide several explicit sufficient conditions for the complete synchronization of Lohe qubit oscillators in terms of the initial condition and coupling strength. We also show that for identical qubit oscillators, the Lohe model for interacting qubits satisfies an asymptotic completeness property. Our analytical results confirm the numerical results from Lohe (J Phys A Math Theor 43:465301, 2010).
Similar content being viewed by others
References
Acebron, J.A., Bonilla, L.L., Pérez Vicente, C.J.P., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005)
Buck, J., Buck, E.: Biology of synchronous flashing of fireflies. Nature 211, 562 (1966)
Choi, Y., Ha, S.-Y., Jung, S., Kim, Y.: Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model. Physica D 241, 735–754 (2012)
Choi, Y.-P., Ha, S.-Y., Yun, S.-B.: Complete synchronization of Kuramoto oscillators with finite inertia. Physica D 240, 32–44 (2011)
Chopra, N., Spong, M.W.: On exponential synchronization of Kuramoto oscillators. IEEE Trans. Autom. Control 54, 353–357 (2009)
Daniels, B.C., Dissanayake, S.T., Trees, B.R.: Synchronization of coupled rotators: Josephson junction ladders and the locally coupled Kuramoto model. Phys. Rev. E 67, 026216 (2003)
Dörfler, F., Bullo, F.: On the critical coupling for Kuramoto oscillators. SIAM J. Appl. Dyn. Syst. 10, 1070–1099 (2011)
Giorgi, G.L., Galve, F., Manzano, G., Colet, P., Zambrini, R.: Quantum correlations and mutual synchronization. Phys. Rev. A 85, 052101 (2012)
Goychuk, I., Casado-Pascual, J., Morillo, M., Lehmann, J., Hänggi, P.: Quantum stochastic synchronization. Phys. Rev. Lett. 97, 210601 (2006)
Ha, S.-Y., Ha, T., Kim, J.-H.: On the complete synchronization of the Kuramoto phase model. Physica D 239, 1692–1700 (2010)
Jadbabaie, A., Motee, N., Barahona, M.: On the stability of the Kuramoto model of coupled nonlinear oscillators. In: Proceedings on American Control Conference, Boston, MA (2004)
Kimble, H.J.: The quantum internet. Nature 453, 1023–1030 (2008)
Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Springer, Berlin (1984)
Kuramoto, Y.: International symposium on mathematical problems in mathematical physics. Lecture Notes Theor. Phys. 30, 420 (1975)
Lohe, M.A.: Quantum synchronization over quantum networks. J. Phys. A Math. Theor. 43, 465301 (2010)
Lohe, M.A.: Non-abelian Kuramoto model and synchronization. J. Phys. A Math. Theor. 42, 395101–395126 (2009)
Machida, M., Kano, T., Yamada, S., Okumura, M., Imamura, T., Koyama, T.: Quantum synchronization effects in intrinsic Josephson junctions. Physica C 468, 689–694 (2008)
Peskin, C.S.: Mathematical Aspects of Heart Physiology. Courant Institute of Mathematical Sciences, New York (1975)
Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: a universal concept in nonlinear sciences. Cambridge University Press, Cambridge (2001)
Vinokur, V.M., Baturina, T.I., Fistul, M.V., Mironov, A.Y., Baklanov, M.R., Strunk, C.: Superinsulator and quantum synchronization. Nature 452, 613–616 (2008)
Watanabe, S., Swift, J.W.: Stability of periodic solutions in series arrays of Josephson junctions with internal capacitance. J. Nonlinear Sci. 7, 503–536 (1997)
Watanabe, S., Strogatz, S.H.: Constants of motion for superconducting Josephson arrays. Physica D 74, 197–253 (1994)
Wiesenfeld, K., Colet, R., Strogatz, S.H.: Frequency locking in Josephson arrays: connection with the Kuramoto model. Phys. Rev. E 57, 1563–1569 (1988)
Wiesenfeld, K., Colet, R., Strogatz, S.H.: Synchronization transitions in a disordered Josephson series arrays. Phys. Rev. Lett. 76, 404–407 (1996)
Wiesenfeld, K., Swift, J.W.: Averaged equations for Josephson junction series arrays. Phys. Rev. E 51, 1020–1025 (1995)
Winfree, A.T.: The Geometry of Biological Time. Springer, New York (1980)
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)
Zhirov, O.V., Shepelyansky, D.L.: Quantum synchronization. Eur. Phys. J. D. 38, 375–379 (2006)
Acknowledgments
The research of S.-H. Choi was supported by BK21 Plus-KAIST. The research of S.-Y. Ha was supported by NRF grant (2014R1A2A205002096).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Paul Newton.
Rights and permissions
About this article
Cite this article
Choi, SH., Ha, SY. Emergent Behaviors of Quantum Lohe Oscillators with All-to-All Coupling. J Nonlinear Sci 25, 1257–1283 (2015). https://doi.org/10.1007/s00332-015-9255-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-015-9255-8