October  2011, 4(5): 1095-1105. doi: 10.3934/dcdss.2011.4.1095

The dynamics of the kink in curved large area Josephson junction

1. 

Institute of Physics UP, Podchorążych 2, 30-084 Cracow, Poland

Received  August 2009 Revised  December 2009 Published  December 2010

A formalism that allows description of the kink motion in an arbitrarily curved large area Josephson junction is proposed. A general formula for the lagrangian density that describes the curved Josephson junction, in small curvature regime, is obtained. Examples of propagation of the kink along the curved Josephson junction are considered.
Citation: Tomasz Dobrowolski. The dynamics of the kink in curved large area Josephson junction. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1095-1105. doi: 10.3934/dcdss.2011.4.1095
References:
[1]

M. J. Ablowitz and P. A. Clarkson, "Solitons, Nonlinear Evolution Equations and Inverse Scattering," London Mathematical Society Lecture Note Series, 149, Cambridge Univ. Press, 1991.

[2]

P. W. Anderson and J. M. Rowell, Probable observation of the Josephson superconducting tunneling effect, Phys. Rev. Lett., 10 (1963), 230-232. doi: 10.1103/PhysRevLett.10.230.

[3]

H. Arodź and R. Pełka, Evolution of interfaces and expansion in width, Phys. Rev. E, 62 (2000), 6749-6759. doi: 10.1103/PhysRevE.62.6749.

[4]

O. Babelon, D. Bernard and M. Talon, "Introduction to Classical Integrable Systems," Cambridge Monographs on Mathematical Physics, Cambridge Univ. Press, 2003.

[5]

A. Barone and G. Paterno, "Physics and Applications of the Josephson Effect," Wiley, New York, 1982. doi: 10.1002/352760278X.

[6]

T. Dobrowolski, Construction of curved domain walls, Phys. Rev. E, 77 (2008), 056608 (5 pages).

[7]

A. Ekert and R. Jozsa, Quantum computation and Shor's factoring algorithm, Rev. Mod. Phys., 68 (1996), 733-753. doi: 10.1103/RevModPhys.68.733.

[8]

J. C. Fernandez, M. J. Goupil, O. Legrand and G. Reinisch, Relativistic dynamics of sine-Gordon solitons trapped in confining potentials, Phys. Rev. B, 34 (1986), 6207-6213. doi: 10.1103/PhysRevB.34.6207.

[9]

L. A. Ferreira, B. Piette and W. J. Zakrzewski, Wobbles and other kink-breather solutions of the sine-Gordon model, Phys. Rev. E, 77 (2008), 036613 (9 pages), preprint arXiv:0708.1088.

[10]

J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo and J. E. Lukens, Quantum superposition of distinct macroscopic states, Nature, 406 (2000), 43-46. doi: 10.1038/35017505.

[11]

C. Gorria, Yu. B. Gaididei, M. P. Soerensen, P. L. Christiansen and J. G. Caputo, Kink propagation and trapping in a two-dimensional curved Josephson junction, Phys. Rev. B, 69 (2004), 134506 (10 pages), preprint, arXiv:nlin/0309001.

[12]

B. D. Josephson, Possible new effects in superconductive tunnelling, Phys. Lett., 1 (1962), 251-253. doi: 10.1016/0031-9163(62)91369-0.

[13]

B. D. Josephson, Supercurrents through barriers, Adv. Phys., 14 (1965), 419-451. doi: 10.1080/00018736500101091.

[14]

A. Kemp, A. Wallraff, A. V. Ustinov, Josephson vortex qubit: Design, preparation and read-out, Phys. Stat. Sol., B 233 (2002), 472-481. doi: 10.1002/1521-3951(200210)233:3<472::AID-PSSB472>3.0.CO;2-J.

[15]

K. K. Kobayashi and M. Izutsu, Exact solution on the n-dimensional sine-Gordon equation, J. Phys. Soc. Japan, 41 (1976), 1091-1092. doi: 10.1143/JPSJ.41.1091.

[16]

G. Leibbrandt, New exact solutions of the classical sine-Gordon equation in 2 + 1 and 3 + 1 dimensions, Phys. Rev. Lett., 41 (1978), 435-438. doi: 10.1103/PhysRevLett.41.435.

[17]

B. A. Malomed, Dynamics of quasi-one-dimensional kinks in the two-dimensional sine-Gordon model, Physica D, 52 (1991), 157-170. doi: 10.1016/0167-2789(91)90118-S.

[18]

Y. Nakamura, Yu. A. Pashkin and J. S. Tsai, Coherent control of macroscopic quantum states in a single-Cooperpair box, Nature, 398 (1999), 786-788, preprint arXiv:cond-mat/9904003.

[19]

M. J. Rice, Physical dynamics of solitons, Phys. Rev. B, 28 (1983), 3587-3589. doi: 10.1103/PhysRevB.28.3587.

[20]

E. Turlot, D. Esteve, C. Urbina and M. Devoret, Dynamical isoperimeter pattern in the square sine-Gordon system, Phys. Rev. B, 42 (1990), 8418-8425. doi: 10.1103/PhysRevB.42.8418.

[21]

C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd and J. E. Mooij, Quantum superposition of macroscopic persistent-current states, Science, 290 (2000), 773-777. doi: 10.1126/science.290.5492.773.

[22]

J. Zagrodziński, Particular solutions of the sine-Gordon equation in 2 + 1 dimensions, Phys. Lett. A, 72 (1979), 284-286. doi: 10.1016/0375-9601(79)90469-9.

show all references

References:
[1]

M. J. Ablowitz and P. A. Clarkson, "Solitons, Nonlinear Evolution Equations and Inverse Scattering," London Mathematical Society Lecture Note Series, 149, Cambridge Univ. Press, 1991.

[2]

P. W. Anderson and J. M. Rowell, Probable observation of the Josephson superconducting tunneling effect, Phys. Rev. Lett., 10 (1963), 230-232. doi: 10.1103/PhysRevLett.10.230.

[3]

H. Arodź and R. Pełka, Evolution of interfaces and expansion in width, Phys. Rev. E, 62 (2000), 6749-6759. doi: 10.1103/PhysRevE.62.6749.

[4]

O. Babelon, D. Bernard and M. Talon, "Introduction to Classical Integrable Systems," Cambridge Monographs on Mathematical Physics, Cambridge Univ. Press, 2003.

[5]

A. Barone and G. Paterno, "Physics and Applications of the Josephson Effect," Wiley, New York, 1982. doi: 10.1002/352760278X.

[6]

T. Dobrowolski, Construction of curved domain walls, Phys. Rev. E, 77 (2008), 056608 (5 pages).

[7]

A. Ekert and R. Jozsa, Quantum computation and Shor's factoring algorithm, Rev. Mod. Phys., 68 (1996), 733-753. doi: 10.1103/RevModPhys.68.733.

[8]

J. C. Fernandez, M. J. Goupil, O. Legrand and G. Reinisch, Relativistic dynamics of sine-Gordon solitons trapped in confining potentials, Phys. Rev. B, 34 (1986), 6207-6213. doi: 10.1103/PhysRevB.34.6207.

[9]

L. A. Ferreira, B. Piette and W. J. Zakrzewski, Wobbles and other kink-breather solutions of the sine-Gordon model, Phys. Rev. E, 77 (2008), 036613 (9 pages), preprint arXiv:0708.1088.

[10]

J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo and J. E. Lukens, Quantum superposition of distinct macroscopic states, Nature, 406 (2000), 43-46. doi: 10.1038/35017505.

[11]

C. Gorria, Yu. B. Gaididei, M. P. Soerensen, P. L. Christiansen and J. G. Caputo, Kink propagation and trapping in a two-dimensional curved Josephson junction, Phys. Rev. B, 69 (2004), 134506 (10 pages), preprint, arXiv:nlin/0309001.

[12]

B. D. Josephson, Possible new effects in superconductive tunnelling, Phys. Lett., 1 (1962), 251-253. doi: 10.1016/0031-9163(62)91369-0.

[13]

B. D. Josephson, Supercurrents through barriers, Adv. Phys., 14 (1965), 419-451. doi: 10.1080/00018736500101091.

[14]

A. Kemp, A. Wallraff, A. V. Ustinov, Josephson vortex qubit: Design, preparation and read-out, Phys. Stat. Sol., B 233 (2002), 472-481. doi: 10.1002/1521-3951(200210)233:3<472::AID-PSSB472>3.0.CO;2-J.

[15]

K. K. Kobayashi and M. Izutsu, Exact solution on the n-dimensional sine-Gordon equation, J. Phys. Soc. Japan, 41 (1976), 1091-1092. doi: 10.1143/JPSJ.41.1091.

[16]

G. Leibbrandt, New exact solutions of the classical sine-Gordon equation in 2 + 1 and 3 + 1 dimensions, Phys. Rev. Lett., 41 (1978), 435-438. doi: 10.1103/PhysRevLett.41.435.

[17]

B. A. Malomed, Dynamics of quasi-one-dimensional kinks in the two-dimensional sine-Gordon model, Physica D, 52 (1991), 157-170. doi: 10.1016/0167-2789(91)90118-S.

[18]

Y. Nakamura, Yu. A. Pashkin and J. S. Tsai, Coherent control of macroscopic quantum states in a single-Cooperpair box, Nature, 398 (1999), 786-788, preprint arXiv:cond-mat/9904003.

[19]

M. J. Rice, Physical dynamics of solitons, Phys. Rev. B, 28 (1983), 3587-3589. doi: 10.1103/PhysRevB.28.3587.

[20]

E. Turlot, D. Esteve, C. Urbina and M. Devoret, Dynamical isoperimeter pattern in the square sine-Gordon system, Phys. Rev. B, 42 (1990), 8418-8425. doi: 10.1103/PhysRevB.42.8418.

[21]

C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd and J. E. Mooij, Quantum superposition of macroscopic persistent-current states, Science, 290 (2000), 773-777. doi: 10.1126/science.290.5492.773.

[22]

J. Zagrodziński, Particular solutions of the sine-Gordon equation in 2 + 1 dimensions, Phys. Lett. A, 72 (1979), 284-286. doi: 10.1016/0375-9601(79)90469-9.

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