Dynamical system approach to synchronization of the coupled Schrödinger–Lohe system
Authors:
Hyungjin Huh and Seung-Yeal Ha
Journal:
Quart. Appl. Math. 75 (2017), 555-579
MSC (2010):
Primary 82C10; Secondary 82C22, 35B35
DOI:
https://doi.org/10.1090/qam/1465
Published electronically:
March 6, 2017
MathSciNet review:
3636169
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Abstract: We study wave function synchronization of the Schrödinger–Lohe model, which describes the dynamics of the ensemble of coupled quantum Lohe oscillators with infinite states. To do this, we first derive a coupled system of ordinary differential equations for the $L^2_x$ inner products between distinct wave functions. For the same one-body potentials, we show that the inner products of two wave functions converge to unity for some restricted class of initial data, so complete wave function synchronization emerges asymptotically when the dynamical system approach is used. Moreover, for the family of one-body potentials consisting of real-value translations of the same base potential, we show that the inner products for a two-oscillator system follow the motion of harmonic oscillators in a small coupling regime, and then as the coupling strength increases, the inner products converge to constant values; this behavior yields convergence toward constant values for the $L^2_x$ differences between distinct wave functions.
References
- J. A. Acebron, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77 (2005), 137–185.
- Neil J. Balmforth and Roberto Sassi, A shocking display of synchrony, Phys. D 143 (2000), no. 1-4, 21–55. Bifurcations, patterns and symmetry. MR 1783383, DOI https://doi.org/10.1016/S0167-2789%2800%2900095-6
- J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature 211 (1966), 562.
- Sun-Ho Choi, Junghee Cho, and Seung-Yeal Ha, Practical quantum synchronization for the Schrödinger-Lohe system, J. Phys. A 49 (2016), no. 20, 205203, 17. MR 3499184, DOI https://doi.org/10.1088/1751-8113/49/20/205203
- Sun-Ho Choi and Seung-Yeal Ha, Quantum synchronization of the Schrödinger-Lohe model, J. Phys. A 47 (2014), no. 35, 355104, 16. MR 3254872, DOI https://doi.org/10.1088/1751-8113/47/35/355104
- Young-Pil Choi, Seung-Yeal Ha, Sungeun Jung, and Yongduck Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Phys. D 241 (2012), no. 7, 735–754. MR 2897541, DOI https://doi.org/10.1016/j.physd.2011.11.011
- L.-Y. Duan, B. Wang and H. J. Kimble, Robust quantum gates on neutral atoms with cavity-assisted photon scattering. Phys. Rev. A 72 (2005), 032333.
- C. L. Evans, Partial differential equations. Second Edition, Graduate Studies in Mathematics, vol 19 Amer. Math. Soc., Providence, RI, 2010.
- S.-Y. Ha, H. Kim and S. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime. Commun. Math. Sci. 14 (2016), 1073–1091.
- S.-Y. Ha, S, Noh and J. Park, Interplay of inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators. To appear in Analysis and Applications.
- Seung-Yeal Ha, Se Eun Noh, and Jinyeong Park, Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics, Netw. Heterog. Media 10 (2015), no. 4, 787–807. MR 3411583, DOI https://doi.org/10.3934/nhm.2015.10.787
- M. R. Hush, W. Li, S. Genway, I. Lesanovsky and A. Armour, Spin correlations as a probe of quantum synchronization in trapped-ion phonon lasers. Phys. Re. A 91 (2015), 061401.
- I. Goychuk, J. Casado-Pascual, M. Morillo, J. Lehmann, and P. Hänggi, Quantum stochastic synchronization, Phys. Rev. Lett. 97 (2006), 210601.
- G. L. Giorgi, F. Galve, G. Manzano, P. Colet and R. Zambrini, Quantum correlations and mutual synchronization. Phys. Rev. A 85 (2012), 052101.
- H. J. Kimble, The quantum internet. Nature 453 (2008), 1023–1030.
- Y. Kuramoto, Chemical oscillations, waves, and turbulence, Springer Series in Synergetics, vol. 19, Springer-Verlag, Berlin, 1984. MR 762432
- Yoshiki Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics (Kyoto Univ., Kyoto, 1975) Springer, Berlin, 1975, pp. 420–422. Lecture Notes in Phys., 39. MR 0676492
- T. E. Lee and H. R. Sadeghpour, Quantum synchronization of quantum van der Pol oscillators with trapped ions. Phys. Rev. Lett. 111 (2013), 234101.
- M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A 43 (2010), no. 46, 465301, 20. MR 2735225, DOI https://doi.org/10.1088/1751-8113/43/46/465301
- M. A. Lohe, Non-abelian Kuramoto models and synchronization, J. Phys. A 42 (2009), no. 39, 395101, 25. MR 2539317, DOI https://doi.org/10.1088/1751-8113/42/39/395101
- M. Ludwig and F. Marquardt, Quantum many-body dynamics in optomechanical arrays. Phys. Rev. Lett. 111 (2013), 073603.
- M. Machida, T. Kano, S. Yamada, M. Okumura, T. Imamura and T. Koyama, Quantum synchronization effects in intrinsic Josephson junctions. Physica C 468 (2008), 689–694.
- A. Mari, A. Farace, N. Didier, V. Giovannetti and R. Fazio, Measures of quantum synchronization in continuous variable systems. Phys. Rev. Lett. 111 (2013), 103605.
- Charles S. Peskin, Mathematical aspects of heart physiology, Courant Institute of Mathematical Sciences, New York University, New York, 1975. Notes based on a course given at New York University during the year 1973/74. MR 0414135
- Steven H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Phys. D 143 (2000), no. 1-4, 1–20. Bifurcations, patterns and symmetry. MR 1783382, DOI https://doi.org/10.1016/S0167-2789%2800%2900094-4
- V. M. Vinokur, T. I. Baturina, M. V. Fistul, A. Y. Mironov, M. R. Baklanov and C. Strunk, Superinsulator and quantum synchronization. Nature 452 (2008), 613–616.
- S. Walter, A. Nunnenkamp and C. Bruder, Quantum synchronization of a driven self-sustained oscillator. Phys. Rev. Lett. 112 (2014), 094102.
- A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16 (1967), 15–42.
- 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 (2014), 154101.
- O.V. Zhirov and D. L. Shepelyansky, Quantum synchronization and entanglement of two qubits coupled to a driven dissipative resonator. Phys. Rev. B 80 (2009), 014519.
- O. V. Zhirov and D. L. Shepelyansky, Quantum synchronization. Eur. Phys. J. D 38 (2006), 375–379.
References
- J. A. Acebron, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77 (2005), 137–185.
- N. J. Balmforth and R. Sassi, A shocking display of synchrony, Phys. D 143 (2000), no. 1-4, 21–55. Bifurcations, patterns and symmetry. MR 1783383, DOI https://doi.org/10.1016/S0167-2789%2800%2900095-6
- J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature 211 (1966), 562.
- S.-H. Choi, J. Cho, and S.-Y. Ha, Practical quantum synchronization for the Schrödinger-Lohe system, J. Phys. A 49 (2016), no. 20, 205203, 17. MR 3499184, DOI https://doi.org/10.1088/1751-8113/49/20/205203
- S.-H. Choi and S.-Y. Ha, Quantum synchronization of the Schrödinger-Lohe model, J. Phys. A 47 (2014), no. 35, 355104, 16. MR 3254872, DOI https://doi.org/10.1088/1751-8113/47/35/355104
- Y.-P. Choi, S.-Y. Ha, S. Jung, and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Phys. D 241 (2012), no. 7, 735–754. MR 2897541, DOI https://doi.org/10.1016/j.physd.2011.11.011
- L.-Y. Duan, B. Wang and H. J. Kimble, Robust quantum gates on neutral atoms with cavity-assisted photon scattering. Phys. Rev. A 72 (2005), 032333.
- C. L. Evans, Partial differential equations. Second Edition, Graduate Studies in Mathematics, vol 19 Amer. Math. Soc., Providence, RI, 2010.
- S.-Y. Ha, H. Kim and S. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime. Commun. Math. Sci. 14 (2016), 1073–1091.
- S.-Y. Ha, S, Noh and J. Park, Interplay of inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators. To appear in Analysis and Applications.
- S.-Y. Ha, S. E. Noh, and J. Park, Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics, Netw. Heterog. Media 10 (2015), no. 4, 787–807. MR 3411583, DOI https://doi.org/10.3934/nhm.2015.10.787
- M. R. Hush, W. Li, S. Genway, I. Lesanovsky and A. Armour, Spin correlations as a probe of quantum synchronization in trapped-ion phonon lasers. Phys. Re. A 91 (2015), 061401.
- I. Goychuk, J. Casado-Pascual, M. Morillo, J. Lehmann, and P. Hänggi, Quantum stochastic synchronization, Phys. Rev. Lett. 97 (2006), 210601.
- G. L. Giorgi, F. Galve, G. Manzano, P. Colet and R. Zambrini, Quantum correlations and mutual synchronization. Phys. Rev. A 85 (2012), 052101.
- H. J. Kimble, The quantum internet. Nature 453 (2008), 1023–1030.
- Y. Kuramoto, Chemical oscillations, waves, and turbulence, Springer Series in Synergetics, vol. 19, Springer-Verlag, Berlin, 1984. MR 762432
- Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics (Kyoto Univ., Kyoto, 1975) Springer, Berlin, 1975, pp. 420–422. Lecture Notes in Phys., 39. MR 0676492
- T. E. Lee and H. R. Sadeghpour, Quantum synchronization of quantum van der Pol oscillators with trapped ions. Phys. Rev. Lett. 111 (2013), 234101.
- M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A 43 (2010), no. 46, 465301, 20. MR 2735225, DOI https://doi.org/10.1088/1751-8113/43/46/465301
- M. A. Lohe, Non-abelian Kuramoto models and synchronization, J. Phys. A 42 (2009), no. 39, 395101, 25. MR 2539317, DOI https://doi.org/10.1088/1751-8113/42/39/395101
- M. Ludwig and F. Marquardt, Quantum many-body dynamics in optomechanical arrays. Phys. Rev. Lett. 111 (2013), 073603.
- M. Machida, T. Kano, S. Yamada, M. Okumura, T. Imamura and T. Koyama, Quantum synchronization effects in intrinsic Josephson junctions. Physica C 468 (2008), 689–694.
- A. Mari, A. Farace, N. Didier, V. Giovannetti and R. Fazio, Measures of quantum synchronization in continuous variable systems. Phys. Rev. Lett. 111 (2013), 103605.
- C. S. Peskin, Mathematical aspects of heart physiology, Courant Institute of Mathematical Sciences, New York University, New York, 1975. Notes based on a course given at New York University during the year 1973/74. MR 0414135
- S. H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Phys. D 143 (2000), no. 1-4, 1–20. Bifurcations, patterns and symmetry. MR 1783382, DOI https://doi.org/10.1016/S0167-2789%2800%2900094-4
- V. M. Vinokur, T. I. Baturina, M. V. Fistul, A. Y. Mironov, M. R. Baklanov and C. Strunk, Superinsulator and quantum synchronization. Nature 452 (2008), 613–616.
- S. Walter, A. Nunnenkamp and C. Bruder, Quantum synchronization of a driven self-sustained oscillator. Phys. Rev. Lett. 112 (2014), 094102.
- A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16 (1967), 15–42.
- 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 (2014), 154101.
- O.V. Zhirov and D. L. Shepelyansky, Quantum synchronization and entanglement of two qubits coupled to a driven dissipative resonator. Phys. Rev. B 80 (2009), 014519.
- O. V. Zhirov and D. L. Shepelyansky, Quantum synchronization. Eur. Phys. J. D 38 (2006), 375–379.
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Additional Information
Hyungjin Huh
Affiliation:
Department of Mathematics, Chung-Ang University, Seoul 156-756, Republic of Korea
MR Author ID:
725994
Email:
huh@cau.ac.kr
Seung-Yeal Ha
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea – and – Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Korea
MR Author ID:
684438
Email:
syha@snu.ac.kr
Keywords:
Complete synchronization,
quantum synchronization,
Schrödinger–Lohe model
Received by editor(s):
February 6, 2017
Published electronically:
March 6, 2017
Additional Notes:
The first author was supported in part by the Basic Science Research Program through the NRF funded by the Ministry of Education (2014R1A1A2053747)
The work of the second author was supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03
Article copyright:
© Copyright 2017
Brown University
