Analytical approximations to the dynamics of an array of coupled DC SQUIDs

Regular Article

Abstract

Coupled dynamical systems that operate near the onset of a bifurcation can lead, under certain conditions, to strong signal amplification effects. Over the past years we have studied this generic feature on a wide range of systems, including: magnetic and electric fields sensors, gyroscopic devices, and arrays of loops of superconducting quantum interference devices, also known as SQUIDs. In this work, we consider an array of SQUID loops connected in series as a case study to derive asymptotic analytical approximations to the exact solutions through perturbation analysis. Two approaches are considered. First, a straightforward expansion in which the non-linear parameter related to the inductance of the DC SQUID is treated as the small perturbation parameter. Second, a more accurate procedure that considers the SQUID phase dynamics as non-uniform motion on a circle. This second procedure is readily extended to the series array and it could serve as a mathematical framework to find approximate solutions to related complex systems with high-dimensionality. To the best of our knowledge, an approximate analytical solutions to an array of SQUIDs has not been reported yet in the literature.

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    C.J. Poole, Handbook of Superconductivity (Academic Press, 2000)Google Scholar
  2. 2.
    J. Clarke, Scientific American 271, 46 (1994)CrossRefGoogle Scholar
  3. 3.
    D.V. Delft, P. Kes, J. Overweg, J. Zaanen, in 100 Years of Superconductivity, Leiden, 8 April 2011 Google Scholar
  4. 4.
    R. Kleiner, D. Koelle, F. Ludwig, J. Clarke, Proc. IEEE 92, 1534 (2004)CrossRefGoogle Scholar
  5. 5.
    R.C. Jaklevic, J. Lambe, A.H. Silver, J.E. Mercereau, Phys. Rev. Lett. 12, 159 (1964)ADSCrossRefGoogle Scholar
  6. 6.
    F. London, Superfluids (Wiley, New York, 1950)Google Scholar
  7. 7.
    B. Deaver Jr., W. Fairbank, Phys. Rev. Lett. 7, 43 (1961)ADSCrossRefGoogle Scholar
  8. 8.
    R. Doll, M. Näbauer, Phys. Rev. Lett. 7, 51 (1961)ADSCrossRefGoogle Scholar
  9. 9.
    100 Years of Superconductivity, edited by H. Rogalla, P. Kes (CRC Press, 2012)Google Scholar
  10. 10.
    R.L. Fagaly, Rev. Sci. Instrum. 77, 101101 (2006)ADSCrossRefGoogle Scholar
  11. 11.
    L.E. Fong, J.R. Holzer, K.K. McBride, E.A. Lima, F. Baudenbacher, M. Radparvar, Rev. Sci. Instrum. 76, 053703 (2005)ADSCrossRefGoogle Scholar
  12. 12.
    O. Hahneiser, S. Kohlsmann, M. Hetscher, K.D. Kramer, Bioelectrochemistry and Bioenergetics 37, 51 (1995)CrossRefGoogle Scholar
  13. 13.
    Y. Machitani, N. Kasai, Y. Fujinawa, H. Iitaka, N. Shirai, Y. Hatsukade, K. Nomura, K. Sugiura, A. Ishiyama, T. Nemoto, IEEE Trans. Appl. Supercond. 13, 763 (2003)CrossRefGoogle Scholar
  14. 14.
    P. Schmidt, D. Clark, K.E. Leslie, M. Bick, D.L. Tilbrook, C.P. Foley, Exploration Geophysics 35, 297 (2004)CrossRefGoogle Scholar
  15. 15.
    A. Chwala, R. Stolz, R. IJsselsteijn, F. Bauer, V. Zakosarenko, U. Hubner, H. Meyer, M. Meyer, SEG Technical Program Expanded Abstracts 29, 779 (2010)CrossRefGoogle Scholar
  16. 16.
    M. Espy, S. Baguisa, D. Dunkerley, P. Magnelind, A. Matlashov, T. Owens, H. Sandin, I. Savukov, L. Schultz, A. Urbaitis, P. Volegov, IEEE Trans. Appl. Supercond. 21, 530 (2011)ADSCrossRefGoogle Scholar
  17. 17.
    R. Bradley, J. Clarke, D. Kinion, L. Rosenberg, K. van Bibber, S. Matsuki, M. Mueck, P. Sikivie, Rev. Mod. Phys. 75, 777 (2003)ADSCrossRefGoogle Scholar
  18. 18.
    D.G. Aronson, M. Golubitsky, M. Krupa, Nonlinearity 4, 861 (1991)ADSCrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    M. Inchiosa, A. Bulsara, K. Wiesenfeld, L. Gammaitoni, Phys. Lett. A 252, 20 (1999)ADSCrossRefGoogle Scholar
  20. 20.
    M. Inchiosa, V. In, A. Bulsara, K. Wiesenfeld, T. Heath, M. Choi, Phys. Rev. E 63, 1 (2001)CrossRefGoogle Scholar
  21. 21.
    A. Bulsara, A.K.V. In, P. Longhini, A. Palacios, W. Rappel, J. Acebron, S. Baglio, B. Ando, Phys. Rev. E 70, 036103 (2004)ADSCrossRefGoogle Scholar
  22. 22.
    A. Palacios, J. Aven, P. Longhini, V. In, A. Bulsara, Phys. Rev. E 74, 021122 (2006)ADSCrossRefGoogle Scholar
  23. 23.
    K. Stawiasz, M. Ketchen, IEEE Trans. Appl. Supercond. 3, 1808 (1993)CrossRefGoogle Scholar
  24. 24.
    J. Oppenländer, Ch. Häussler, N. Schopohl, Phys. Rev. B 63, 024511 (2000)CrossRefGoogle Scholar
  25. 25.
    C. Häussler, J. Oppenländer, N. Schopohl, J. Appl. Phys. 89, 1875 (2001)CrossRefGoogle Scholar
  26. 26.
    T. Träuble, J. Oppenländer, C. Häussler, N. Schopohl, Physica C 368, 119 (2002)CrossRefGoogle Scholar
  27. 27.
    J. Oppenländer, C. Häussler, T. Träuble, P. Caputo, J. Tomes, A. Friesch, N. Schopohl, IEEE Trans. Appl. Supercond. 13, 771 (2003)CrossRefGoogle Scholar
  28. 28.
    J. Oppenländer, T. Träuble, C. Häussler, N. Schopohl, IEEE Trans. Appl. Supercond. 11, 1271 (2001)CrossRefGoogle Scholar
  29. 29.
    J. Oppenländer, C. Häussler, A. Friesch, J. Tomes, P. Caputo, T. Träuble, N. Schopohl, IEEE Trans. Appl. Supercond. 15, 936 (2005)CrossRefGoogle Scholar
  30. 30.
    V.K. Kornev, I.I. Soloviev, N.V. Klenov, O.A. Mukhanov, IEEE Trans. Appl. Supercond. 19, 741 (2009)ADSCrossRefGoogle Scholar
  31. 31.
    V.K. Kornev, I.I. Soloviev, J. Oppenländer, C. Häussler, N. Schopohl, Supercond. Sci. Technol. 17, S406 (2004)ADSCrossRefGoogle Scholar
  32. 32.
    V. Schultze, R. IJsselsteijn, H.G. Meyer, J. Oppenländer, C. Häussler, N. Schopohl, IEEE Trans. Appl. Supercond. 13, 775 (2003)CrossRefGoogle Scholar
  33. 33.
    V. Schultze, R. IJsselsteijn, H.G. Meyer, Supercond. Sci. Technol. 19, S411 (2006)ADSCrossRefGoogle Scholar
  34. 34.
    J. Oppenländer, P. Caputo, C. Häussler, T. Träuble, J. Tomes, A. Friesch, N. Schopohl, Appl. Phys. Lett. 83, 969 (2003)ADSCrossRefGoogle Scholar
  35. 35.
    P. Caputo, J. Oppenländer, C. Häussler, J. Tomes, A. Friesch, T. Träuble, N. Schopohl, Appl. Phys. Lett. 85, 1389 (2004)ADSCrossRefGoogle Scholar
  36. 36.
    P. Caputo, J. Tomes, J. Oppenländer, C. Häussler, A. Friesch, T. Träuble, N. Schopohl, IEEE Trans. Appl. Supercond. 15, 1044 (2005)CrossRefGoogle Scholar
  37. 37.
    Y. Polyakov, V. Semenov, S. Tolpygo, IEEE Trans. Appl. Supercond. 21, 724 (2011)ADSCrossRefGoogle Scholar
  38. 38.
    P. Caputo, J. Tomes, J. Oppenländer, C. Häussler, A. Friesch, T. Träuble, N. Schopohl, IEEE Trans. Appl. Supercond. 17, 722 (2006)ADSCrossRefGoogle Scholar
  39. 39.
    V.K. Kornev, I.I. Soloviev, N.V. Klenov, T. Filippov, H. Engseth, O.A. Mukhanov, IEEE Trans. Appl. Supercond. 19, 916 (2009)ADSCrossRefGoogle Scholar
  40. 40.
    V.K. Kornev, I.I. Soloviev, N.V. Klenov, A.V. Sharafiev, O.A. Mukhanov, IEEE Trans. Appl. Supercond. 21, 713 (2011)ADSCrossRefGoogle Scholar
  41. 41.
    P. Caputo, J. Tomes, J. Oppenländer, C. Häussler, A. Friesch, T. Träuble, N. Schopohl, Appl. Phys. Lett. 89, 062507 (2006)ADSCrossRefGoogle Scholar
  42. 42.
    P. Caputo, J. Tomes, J. Oppenländer, Ch. Häussler, A. Friesch, T. Träuble, N. Schopohl, J. Supercond. Novel Magn. 20, 25 (2007)CrossRefGoogle Scholar
  43. 43.
    A. Shadrin, K. Constantinian, G. Ovsyannikov, Tech. Phys. Lett. 33, 192 (2007)ADSCrossRefGoogle Scholar
  44. 44.
    A.K. Kalabukhov, M.L. Chukharkin, A.A. Deleniv, D. Winkler, I.A. Volkov, O.V. Snigirev, J. Commun. Technol. Electron. 53, 934 (2008)CrossRefGoogle Scholar
  45. 45.
    V.K. Kornev, I.I. Soloviev, N.V. Klenov, A.V. Sharafiev, O.A. Mukhanov, Physica C 479, 119 (2012)ADSCrossRefGoogle Scholar
  46. 46.
    K. Wiesenfeld, A. Bulsara, M. Inchiosa, Phys. Rev. B 62, R9232 (2000)ADSCrossRefGoogle Scholar
  47. 47.
    M. Mück, R. McDermott, Supercond. Sci. Technol. 23, 093001 (2010)ADSCrossRefGoogle Scholar
  48. 48.
    C. Hilbert, J. Clarke, J. Low Temp. Phys. 61, 263 (1985)ADSCrossRefGoogle Scholar
  49. 49.
    M. Cyrille, Thin Solid Films 333, 228 (1998)ADSCrossRefGoogle Scholar
  50. 50.
    O. Snigirev, M. Chukharkin, A. Kalabukhov, M. Tarasov, A.A. Deleniv, O.A. Mukhanov, D. Winkler, IEEE Trans. Appl. Supercond. 17, 718 (2007)ADSCrossRefGoogle Scholar
  51. 51.
    V.K. Kornev, I.I. Soloviev, N.V. Klenov, O.A. Mukhanov, IEEE Trans. Appl. Supercond. 17, 569 (2007)ADSCrossRefGoogle Scholar
  52. 52.
    J. Luine, L. Abelson, D. Brundrett, J. Burch, E. Dantsker, K. Hummer, G. Kerber, M. Wire, K. Yokoyama, D. Bowling, M. Neel, S. Hubbell, K. Li, IEEE Trans. Appl. Supercond. 9, 4141 (1999)CrossRefGoogle Scholar
  53. 53.
    A.M.L. Martin, D.R. Bowling, M.M. Neel, Naval Eng. J. 110, 123 (1998)CrossRefGoogle Scholar
  54. 54.
    S.K. Khamas, M.J. Mehler, T.S.M. Maclean, C.E. Gough, Electron. Lett. 24, 460 (1988)ADSCrossRefGoogle Scholar
  55. 55.
    K. Sakuta, Y. Narita, H. Itozaki, Supercond. Sci. Technol. 20, S389 (2007)ADSCrossRefGoogle Scholar
  56. 56.
    T. Lanting, M. Dobbs, H. Spieler, A.T. Lee, Y. Yamamoto, arXiv:0901.1919 [astro-ph.IM] (2009)Google Scholar
  57. 57.
    J. Beyer, D. Drung, Supercond. Sci. Technol. 21, 095012 (2008)ADSCrossRefGoogle Scholar
  58. 58.
    R. De Luca, A. Fedullo, V. Gasanenko, Eur. Phys. J. B 58, 461 (2007)ADSCrossRefGoogle Scholar
  59. 59.
    S. Yukon, DTIC Online (2010), pp. 1–25Google Scholar
  60. 60.
    F. Romeo, R.D. Luca, Phys. Lett. A 328, 330 (2004)ADSCrossRefGoogle Scholar
  61. 61.
    N. Grønbech-Jensen, C. Cosmeli, D. Thompson, M. Cirillo, Phys. Rev. B 67, 224505 (2003)ADSCrossRefGoogle Scholar
  62. 62.
    K. Tsang, R. Mirollo, S. Strogatz, K. Wiesenfeld, Physica D 48, 102 (1991)ADSCrossRefMATHMathSciNetGoogle Scholar
  63. 63.
    M. Bennett, K. Wiesenfeld, Physica D 192, 196 (2004)ADSCrossRefMATHMathSciNetGoogle Scholar
  64. 64.
    K. Wiesenfeld, J. Swift, Phys. Rev. E 51, 1020 (1995)ADSCrossRefGoogle Scholar
  65. 65.
    S. Watanabe, S. Strogatz, Physica D 74, 197 (1994)ADSCrossRefMATHGoogle Scholar
  66. 66.
    J. Swift, S. Strogatz, K. Wiesenfeld, Physica D 55, 239 (1992)ADSCrossRefMATHMathSciNetGoogle Scholar
  67. 67.
    S. Watanabe, J. Swift, J. Nonlinear Sci. 7, 503 (1997)CrossRefMATHMathSciNetGoogle Scholar
  68. 68.
    A. Barone, G. Paterno, Physics and Applications of the Josephson Effect (J. Wiley, New York, 1982)Google Scholar
  69. 69.
    A. Bulsara, J. Acebron, W. Rappel, A. Hibbs, L. Kunstmanas, M. Krupka, Physica A 325, 220 (2003)ADSCrossRefGoogle Scholar
  70. 70.
    J.L. Aven, Networks of coupled SQUID magnetometers, Master’s thesis, San Diego State University, 2006Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.SPAWAR Systems Center PacificSan DiegoUSA
  2. 2.Nonlinear Dynamical Systems Group, Department of MathematicsSan Diego State UniversitySan DiegoUSA

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