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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

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.

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. AcebronL. L. BonillaC. J. P. Pérez VicenteF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

Y.-P. ChoiS.-Y. HaS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

S.-Y. HaH. 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.  Google Scholar

[17]

S.-Y. HaS. 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.  Google Scholar

[18]

S.-Y. HaS. 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.  Google Scholar

[19]

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

[20]

R. IllnerP. 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.  Google Scholar

[21]

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

[22]

P. GérardP. A. MarkowichN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[30]

M. MachidaT. KanoS. YamadaM. OkumuraT. 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.  Google Scholar

[31]

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

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[36]

V. M. VinokurT. I. BaturinaM. V. FistulA. Y. MironovM. R. Baklanov and C. Strunk, Superinsulator and quantum synchronization, Nature, 452 (2008), 613-615.  doi: 10.1038/nature06837.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

show all references

References:
[1]

J. A. AcebronL. L. BonillaC. J. P. Pérez VicenteF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

Y.-P. ChoiS.-Y. HaS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

S.-Y. HaH. 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.  Google Scholar

[17]

S.-Y. HaS. 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.  Google Scholar

[18]

S.-Y. HaS. 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.  Google Scholar

[19]

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

[20]

R. IllnerP. 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.  Google Scholar

[21]

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

[22]

P. GérardP. A. MarkowichN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[30]

M. MachidaT. KanoS. YamadaM. OkumuraT. 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.  Google Scholar

[31]

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

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[36]

V. M. VinokurT. I. BaturinaM. V. FistulA. Y. MironovM. R. Baklanov and C. Strunk, Superinsulator and quantum synchronization, Nature, 452 (2008), 613-615.  doi: 10.1038/nature06837.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

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