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September 2017, 12(3): 403-416. doi: 10.3934/nhm.2017018

The Wigner-Lohe model for quantum synchronization and its emergent dynamics

Paolo Antonelli 1, , Seung-Yeal Ha 2, , Dohyun Kim 2,3, and  Pierangelo Marcati 1,

1. 

Gran Sasso Science Institute, viale F. Crispi, 7, 67100 L'Aquila, Italy
2. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747, Republic of Korea
3. 

Korea Institute for Advanced Study, Hoegiro 87, Seoul, 130-722, Republic of Korea

Received  February 2017 Revised  June 2017 Published  September 2017

Fund Project: The work of P. Antonelli and P. Marcati is partially supported by PRIN grant 2015YCJY3A 003 and by G.N.A.M.P.A. (I.N.d.A.M.), The work of S.-Y. Ha and D. Kim are partially supported by the National Research Foundation of Korea (NRF2017R1A2B2001864)

We present the Wigner-Lohe model for quantum synchronization which can be derived from the Schrödinger-Lohe model using the Wigner formalism. For identical one-body potentials, we provide a priori sufficient framework leading the complete synchronization, in which L2-distances between all wave functions tend to zero asymptotically.

Keywords: Quantum synchronization, emergent behavior, Wigner equation, quantum hydrodynamics.
Mathematics Subject Classification: Primary: 35Q40; Secondary: 82C22, 93D20, 35B40.
Citation: Paolo Antonelli, Seung-Yeal Ha, Dohyun Kim, Pierangelo Marcati. The Wigner-Lohe model for quantum synchronization and its emergent dynamics. Networks & Heterogeneous Media, 2017, 12 (3) : 403-416. doi: 10.3934/nhm.2017018
References:
[1]

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. doi: 10.1103/RevModPhys.77.137.

[2]

P. Antonelli and P. Marcati, The quantum hydrodynamics system in two space dimensions, Arch. Rational Mech. Anal., 203 (2012), 499-527. doi: 10.1007/s00205-011-0454-7.

[3]

P. Antonelli and P. Marcati, On the finite weak solutions to a system in quantum fluid dynamics, Comm. Math. Phys., 287 (2009), 657-686. doi: 10.1007/s00220-008-0632-0.

[4]

P. Antonelli and P. Marcati, Some results on systems for quantum fluids, in Recent Advances in Partial Differential Equations and Applications an International Conference (in honor of H. Beirão da Veiga's 70th birthday), ed. by V. D. Radulescu, A. Sequeira, V. A. Solonnikov. Contemporary Mathematics, vol. 666 (American Mathematical Society, Providence, 2016).
[5]

P. Antonelli and P. Marcati, A model of Synchronization over Quantum Networks, J. Phys. A. , 50 (2017), 315101. doi: 10.1088/1751-8121/aa79c9.

[6]

N. J. Balmforth and R. Sassi, A shocking display of synchrony, Physica D, 143 (2000), 21-55. doi: 10.1016/S0167-2789(00)00095-6.

[7]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564. doi: 10.1038/211562a0.

[8]

C. S. Bohun, R. Illner and P. F. Zweifel, Some remarks on the Wigner transform and the Wigner-Poisson system, Matematiche, 46 (1991), 429-438.

[9]

F. Brezzi and P. A. Markowich, The three-dimensional Wigner-Poisson problem: Existence, uniqueness and appproximation, Math. Methods Appl. Sci., 14 (1991), 35-61. doi: 10.1002/mma.1670140103.

[10]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vol. 10, New York University, Courant Institute of Mathematical Sciences, AMS, 2003. doi: 10.1090/cln/010.

[11]

S. -H. Choi, J. Cho and S. -Y. Ha Practical quantum synchronization for the Schrödinger-Lohe system, J. Phys. A, 49 (2016), 205203, 17pp. doi: 10.1088/1751-8113/49/20/205203.

[12]

S. -H. Choi and S. -Y. Ha, Quantum synchronization of the Schrödinger-Lohe model, J. Phys. A, 47 (2014), 355104, 16pp. doi: 10.1088/1751-8113/47/35/355104.

[13]

Y.-P. Choi, S.-Y. Ha, S. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754. doi: 10.1016/j.physd.2011.11.011.

[14]

L. -M. Duan, B. Wang and H. J. Kimble, Robust quantum gates on neutral atoms with cavityassisted photon scattering. Phys. Rev. A, 72 (2005), 032333. doi: 10.1103/PhysRevA.72.032333.

[15]

S.-Y. Ha and H. Huh, Dynamical system approach to synchronization of the coupled Schrödinger-Lohe system, Quart. Appl. Math., 75 (2017), 555-579. doi: 10.1090/qam/1465.

[16]

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. doi: 10.4310/CMS.2016.v14.n4.a10.

[17]

S.-Y. Ha, S. Noh and J. Park, Interplay of inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators, Analysis and Applications, 15 (2017), 837-861. doi: 10.1142/S0219530516500111.

[18]

S.-Y. Ha, S. Noh and J. Park, Practical synchronization of generalized Kuramoto system with an intrinsic dynamics, Netw. Heterog. Media, 10 (2015), 787-807. doi: 10.3934/nhm.2015.10.787.

[19]

R. Illner, uniqueness and asymptotic behavior of Wigner-Poisson and VlasovPoisson systems: A survey, Transp. Theory Stat. Phys., 26 (1997), 195-207.

[20]

R. Illner, P. F. Zweifel and H. Lange, Global existence, uniqueness and asymptotic behavior of solutions of the Wigner-Poisson and Schrödinger-Poisson systems, Math. Methods Appl. Sci., 17 (1994), 349-376. doi: 10.1002/mma.1670170504.

[21]

I. Gasser, P. A. Markowich and B. Perthame, Dispersion and moment lemmas revisited, J. Diff. Eq., 156 (1999), 254-281. doi: 10.1006/jdeq.1998.3595.

[22]

P. Gérard, P. A. Markowich, N. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.

[23]

I. Goychuk, J. Casado-Pascual, M. Morillo, J. Lehmann and P. Hänggi, , Quantum stochastic synchronization, Phys. Rev. Lett. , 97 (2006), 210601. doi: 10.1103/PhysRevLett.97.210601.

[24]

G. L. Giorgi, F. Galve, G. Manzano, P. Colet and R. Zambrini, Quantum correlations and mutual synchronization, Phys. Rev. A, 85 (2012), 052101. doi: 10.1103/PhysRevA.85.052101.

[25]

H. J. Kimble, The quantum internet, Nature, 453 (2008), 1023-1030. doi: 10.1038/nature07127.

[26]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag. Berlin. 1984. doi: 10.1007/978-3-642-69689-3.

[27]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), p420.
[28]

M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A, 43 (2010), 465301, 20pp. doi: 10.1088/1751-8113/43/46/465301.

[29]

M. A. Lohe, Non-Abelian Kuramoto model and synchronization, J. Phys. A, 42 (2009), 395101, 25pp. doi: 10.1088/1751-8113/42/39/395101.

[30]

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. doi: 10.1016/j.physc.2007.11.081.

[31]

E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Phys., 40 (1927), 322-326. doi: 10.1007/BF01400372.

[32]

P. A. Markowich, On the equivalence of the Schrödinger and the quantum Liouville equation, Math. Methods Appl. Sci., 11 (1989), 459-469. doi: 10.1002/mma.1670110404.

[33]

C. S. Peskin, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York, 1975.

[34]

H. Steinrück, The one-dimensional Wigner-Poisson problem and a relation to the SchrödingerPoisson problem, SIAM J. Math. Anal., 22 (1991), 957-972. doi: 10.1137/0522061.

[35]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20. doi: 10.1016/S0167-2789(00)00094-4.

[36]

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), 613-615. doi: 10.1038/nature06837.

[37]

A.T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. doi: 10.1016/0022-5193(67)90051-3.

[38]

E. P. Wigner, On the quantum correction for thermodynamic equilibrium, Part Ⅰ: Physical Chemistry. Part Ⅱ: Solid State Physics, (1997), 110-120. doi: 10.1007/978-3-642-59033-7_9.

[39]

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. doi: 10.1103/PhysRevB. 80. 014519.

[40]

O. V. Zhirov and D. L. Shepelyansky, Quantum synchronization, Eur. Phys. J. D, 38 (2006), 375-379. doi: 10.1140/epjd/e2006-00011-9.

[41]

P. F. Zweifel, The Wigner transform and the Wigner-Poisson system, Transp. Theory Stat. Phys., 22 (1993), 459-484. doi: 10.1080/00411459308203824.

show all references

References:
[1]

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. doi: 10.1103/RevModPhys.77.137.

[2]

P. Antonelli and P. Marcati, The quantum hydrodynamics system in two space dimensions, Arch. Rational Mech. Anal., 203 (2012), 499-527. doi: 10.1007/s00205-011-0454-7.

[3]

P. Antonelli and P. Marcati, On the finite weak solutions to a system in quantum fluid dynamics, Comm. Math. Phys., 287 (2009), 657-686. doi: 10.1007/s00220-008-0632-0.

[4]

P. Antonelli and P. Marcati, Some results on systems for quantum fluids, in Recent Advances in Partial Differential Equations and Applications an International Conference (in honor of H. Beirão da Veiga's 70th birthday), ed. by V. D. Radulescu, A. Sequeira, V. A. Solonnikov. Contemporary Mathematics, vol. 666 (American Mathematical Society, Providence, 2016).
[5]

P. Antonelli and P. Marcati, A model of Synchronization over Quantum Networks, J. Phys. A. , 50 (2017), 315101. doi: 10.1088/1751-8121/aa79c9.

[6]

N. J. Balmforth and R. Sassi, A shocking display of synchrony, Physica D, 143 (2000), 21-55. doi: 10.1016/S0167-2789(00)00095-6.

[7]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564. doi: 10.1038/211562a0.

[8]

C. S. Bohun, R. Illner and P. F. Zweifel, Some remarks on the Wigner transform and the Wigner-Poisson system, Matematiche, 46 (1991), 429-438.

[9]

F. Brezzi and P. A. Markowich, The three-dimensional Wigner-Poisson problem: Existence, uniqueness and appproximation, Math. Methods Appl. Sci., 14 (1991), 35-61. doi: 10.1002/mma.1670140103.

[10]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vol. 10, New York University, Courant Institute of Mathematical Sciences, AMS, 2003. doi: 10.1090/cln/010.

[11]

S. -H. Choi, J. Cho and S. -Y. Ha Practical quantum synchronization for the Schrödinger-Lohe system, J. Phys. A, 49 (2016), 205203, 17pp. doi: 10.1088/1751-8113/49/20/205203.

[12]

S. -H. Choi and S. -Y. Ha, Quantum synchronization of the Schrödinger-Lohe model, J. Phys. A, 47 (2014), 355104, 16pp. doi: 10.1088/1751-8113/47/35/355104.

[13]

Y.-P. Choi, S.-Y. Ha, S. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754. doi: 10.1016/j.physd.2011.11.011.

[14]

L. -M. Duan, B. Wang and H. J. Kimble, Robust quantum gates on neutral atoms with cavityassisted photon scattering. Phys. Rev. A, 72 (2005), 032333. doi: 10.1103/PhysRevA.72.032333.

[15]

S.-Y. Ha and H. Huh, Dynamical system approach to synchronization of the coupled Schrödinger-Lohe system, Quart. Appl. Math., 75 (2017), 555-579. doi: 10.1090/qam/1465.

[16]

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. doi: 10.4310/CMS.2016.v14.n4.a10.

[17]

S.-Y. Ha, S. Noh and J. Park, Interplay of inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators, Analysis and Applications, 15 (2017), 837-861. doi: 10.1142/S0219530516500111.

[18]

S.-Y. Ha, S. Noh and J. Park, Practical synchronization of generalized Kuramoto system with an intrinsic dynamics, Netw. Heterog. Media, 10 (2015), 787-807. doi: 10.3934/nhm.2015.10.787.

[19]

R. Illner, uniqueness and asymptotic behavior of Wigner-Poisson and VlasovPoisson systems: A survey, Transp. Theory Stat. Phys., 26 (1997), 195-207.

[20]

R. Illner, P. F. Zweifel and H. Lange, Global existence, uniqueness and asymptotic behavior of solutions of the Wigner-Poisson and Schrödinger-Poisson systems, Math. Methods Appl. Sci., 17 (1994), 349-376. doi: 10.1002/mma.1670170504.

[21]

I. Gasser, P. A. Markowich and B. Perthame, Dispersion and moment lemmas revisited, J. Diff. Eq., 156 (1999), 254-281. doi: 10.1006/jdeq.1998.3595.

[22]

P. Gérard, P. A. Markowich, N. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.

[23]

I. Goychuk, J. Casado-Pascual, M. Morillo, J. Lehmann and P. Hänggi, , Quantum stochastic synchronization, Phys. Rev. Lett. , 97 (2006), 210601. doi: 10.1103/PhysRevLett.97.210601.

[24]

G. L. Giorgi, F. Galve, G. Manzano, P. Colet and R. Zambrini, Quantum correlations and mutual synchronization, Phys. Rev. A, 85 (2012), 052101. doi: 10.1103/PhysRevA.85.052101.

[25]

H. J. Kimble, The quantum internet, Nature, 453 (2008), 1023-1030. doi: 10.1038/nature07127.

[26]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag. Berlin. 1984. doi: 10.1007/978-3-642-69689-3.

[27]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), p420.
[28]

M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A, 43 (2010), 465301, 20pp. doi: 10.1088/1751-8113/43/46/465301.

[29]

M. A. Lohe, Non-Abelian Kuramoto model and synchronization, J. Phys. A, 42 (2009), 395101, 25pp. doi: 10.1088/1751-8113/42/39/395101.

[30]

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. doi: 10.1016/j.physc.2007.11.081.

[31]

E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Phys., 40 (1927), 322-326. doi: 10.1007/BF01400372.

[32]

P. A. Markowich, On the equivalence of the Schrödinger and the quantum Liouville equation, Math. Methods Appl. Sci., 11 (1989), 459-469. doi: 10.1002/mma.1670110404.

[33]

C. S. Peskin, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York, 1975.

[34]

H. Steinrück, The one-dimensional Wigner-Poisson problem and a relation to the SchrödingerPoisson problem, SIAM J. Math. Anal., 22 (1991), 957-972. doi: 10.1137/0522061.

[35]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20. doi: 10.1016/S0167-2789(00)00094-4.

[36]

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), 613-615. doi: 10.1038/nature06837.

[37]

A.T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. doi: 10.1016/0022-5193(67)90051-3.

[38]

E. P. Wigner, On the quantum correction for thermodynamic equilibrium, Part Ⅰ: Physical Chemistry. Part Ⅱ: Solid State Physics, (1997), 110-120. doi: 10.1007/978-3-642-59033-7_9.

[39]

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. doi: 10.1103/PhysRevB. 80. 014519.

[40]

O. V. Zhirov and D. L. Shepelyansky, Quantum synchronization, Eur. Phys. J. D, 38 (2006), 375-379. doi: 10.1140/epjd/e2006-00011-9.

[41]

P. F. Zweifel, The Wigner transform and the Wigner-Poisson system, Transp. Theory Stat. Phys., 22 (1993), 459-484. doi: 10.1080/00411459308203824.

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