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.
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References
C.J. Poole, Handbook of Superconductivity (Academic Press, 2000)
J. Clarke, Scientific American 271, 46 (1994)
D.V. Delft, P. Kes, J. Overweg, J. Zaanen, in 100 Years of Superconductivity, Leiden, 8 April 2011
R. Kleiner, D. Koelle, F. Ludwig, J. Clarke, Proc. IEEE 92, 1534 (2004)
R.C. Jaklevic, J. Lambe, A.H. Silver, J.E. Mercereau, Phys. Rev. Lett. 12, 159 (1964)
F. London, Superfluids (Wiley, New York, 1950)
B. Deaver Jr., W. Fairbank, Phys. Rev. Lett. 7, 43 (1961)
R. Doll, M. Näbauer, Phys. Rev. Lett. 7, 51 (1961)
100 Years of Superconductivity, edited by H. Rogalla, P. Kes (CRC Press, 2012)
R.L. Fagaly, Rev. Sci. Instrum. 77, 101101 (2006)
L.E. Fong, J.R. Holzer, K.K. McBride, E.A. Lima, F. Baudenbacher, M. Radparvar, Rev. Sci. Instrum. 76, 053703 (2005)
O. Hahneiser, S. Kohlsmann, M. Hetscher, K.D. Kramer, Bioelectrochemistry and Bioenergetics 37, 51 (1995)
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)
P. Schmidt, D. Clark, K.E. Leslie, M. Bick, D.L. Tilbrook, C.P. Foley, Exploration Geophysics 35, 297 (2004)
A. Chwala, R. Stolz, R. IJsselsteijn, F. Bauer, V. Zakosarenko, U. Hubner, H. Meyer, M. Meyer, SEG Technical Program Expanded Abstracts 29, 779 (2010)
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)
R. Bradley, J. Clarke, D. Kinion, L. Rosenberg, K. van Bibber, S. Matsuki, M. Mueck, P. Sikivie, Rev. Mod. Phys. 75, 777 (2003)
D.G. Aronson, M. Golubitsky, M. Krupa, Nonlinearity 4, 861 (1991)
M. Inchiosa, A. Bulsara, K. Wiesenfeld, L. Gammaitoni, Phys. Lett. A 252, 20 (1999)
M. Inchiosa, V. In, A. Bulsara, K. Wiesenfeld, T. Heath, M. Choi, Phys. Rev. E 63, 1 (2001)
A. Bulsara, A.K.V. In, P. Longhini, A. Palacios, W. Rappel, J. Acebron, S. Baglio, B. Ando, Phys. Rev. E 70, 036103 (2004)
A. Palacios, J. Aven, P. Longhini, V. In, A. Bulsara, Phys. Rev. E 74, 021122 (2006)
K. Stawiasz, M. Ketchen, IEEE Trans. Appl. Supercond. 3, 1808 (1993)
J. Oppenländer, Ch. Häussler, N. Schopohl, Phys. Rev. B 63, 024511 (2000)
C. Häussler, J. Oppenländer, N. Schopohl, J. Appl. Phys. 89, 1875 (2001)
T. Träuble, J. Oppenländer, C. Häussler, N. Schopohl, Physica C 368, 119 (2002)
J. Oppenländer, C. Häussler, T. Träuble, P. Caputo, J. Tomes, A. Friesch, N. Schopohl, IEEE Trans. Appl. Supercond. 13, 771 (2003)
J. Oppenländer, T. Träuble, C. Häussler, N. Schopohl, IEEE Trans. Appl. Supercond. 11, 1271 (2001)
J. Oppenländer, C. Häussler, A. Friesch, J. Tomes, P. Caputo, T. Träuble, N. Schopohl, IEEE Trans. Appl. Supercond. 15, 936 (2005)
V.K. Kornev, I.I. Soloviev, N.V. Klenov, O.A. Mukhanov, IEEE Trans. Appl. Supercond. 19, 741 (2009)
V.K. Kornev, I.I. Soloviev, J. Oppenländer, C. Häussler, N. Schopohl, Supercond. Sci. Technol. 17, S406 (2004)
V. Schultze, R. IJsselsteijn, H.G. Meyer, J. Oppenländer, C. Häussler, N. Schopohl, IEEE Trans. Appl. Supercond. 13, 775 (2003)
V. Schultze, R. IJsselsteijn, H.G. Meyer, Supercond. Sci. Technol. 19, S411 (2006)
J. Oppenländer, P. Caputo, C. Häussler, T. Träuble, J. Tomes, A. Friesch, N. Schopohl, Appl. Phys. Lett. 83, 969 (2003)
P. Caputo, J. Oppenländer, C. Häussler, J. Tomes, A. Friesch, T. Träuble, N. Schopohl, Appl. Phys. Lett. 85, 1389 (2004)
P. Caputo, J. Tomes, J. Oppenländer, C. Häussler, A. Friesch, T. Träuble, N. Schopohl, IEEE Trans. Appl. Supercond. 15, 1044 (2005)
Y. Polyakov, V. Semenov, S. Tolpygo, IEEE Trans. Appl. Supercond. 21, 724 (2011)
P. Caputo, J. Tomes, J. Oppenländer, C. Häussler, A. Friesch, T. Träuble, N. Schopohl, IEEE Trans. Appl. Supercond. 17, 722 (2006)
V.K. Kornev, I.I. Soloviev, N.V. Klenov, T. Filippov, H. Engseth, O.A. Mukhanov, IEEE Trans. Appl. Supercond. 19, 916 (2009)
V.K. Kornev, I.I. Soloviev, N.V. Klenov, A.V. Sharafiev, O.A. Mukhanov, IEEE Trans. Appl. Supercond. 21, 713 (2011)
P. Caputo, J. Tomes, J. Oppenländer, C. Häussler, A. Friesch, T. Träuble, N. Schopohl, Appl. Phys. Lett. 89, 062507 (2006)
P. Caputo, J. Tomes, J. Oppenländer, Ch. Häussler, A. Friesch, T. Träuble, N. Schopohl, J. Supercond. Novel Magn. 20, 25 (2007)
A. Shadrin, K. Constantinian, G. Ovsyannikov, Tech. Phys. Lett. 33, 192 (2007)
A.K. Kalabukhov, M.L. Chukharkin, A.A. Deleniv, D. Winkler, I.A. Volkov, O.V. Snigirev, J. Commun. Technol. Electron. 53, 934 (2008)
V.K. Kornev, I.I. Soloviev, N.V. Klenov, A.V. Sharafiev, O.A. Mukhanov, Physica C 479, 119 (2012)
K. Wiesenfeld, A. Bulsara, M. Inchiosa, Phys. Rev. B 62, R9232 (2000)
M. Mück, R. McDermott, Supercond. Sci. Technol. 23, 093001 (2010)
C. Hilbert, J. Clarke, J. Low Temp. Phys. 61, 263 (1985)
M. Cyrille, Thin Solid Films 333, 228 (1998)
O. Snigirev, M. Chukharkin, A. Kalabukhov, M. Tarasov, A.A. Deleniv, O.A. Mukhanov, D. Winkler, IEEE Trans. Appl. Supercond. 17, 718 (2007)
V.K. Kornev, I.I. Soloviev, N.V. Klenov, O.A. Mukhanov, IEEE Trans. Appl. Supercond. 17, 569 (2007)
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)
A.M.L. Martin, D.R. Bowling, M.M. Neel, Naval Eng. J. 110, 123 (1998)
S.K. Khamas, M.J. Mehler, T.S.M. Maclean, C.E. Gough, Electron. Lett. 24, 460 (1988)
K. Sakuta, Y. Narita, H. Itozaki, Supercond. Sci. Technol. 20, S389 (2007)
T. Lanting, M. Dobbs, H. Spieler, A.T. Lee, Y. Yamamoto, arXiv:0901.1919 [astro-ph.IM] (2009)
J. Beyer, D. Drung, Supercond. Sci. Technol. 21, 095012 (2008)
R. De Luca, A. Fedullo, V. Gasanenko, Eur. Phys. J. B 58, 461 (2007)
S. Yukon, DTIC Online (2010), pp. 1–25
F. Romeo, R.D. Luca, Phys. Lett. A 328, 330 (2004)
N. Grønbech-Jensen, C. Cosmeli, D. Thompson, M. Cirillo, Phys. Rev. B 67, 224505 (2003)
K. Tsang, R. Mirollo, S. Strogatz, K. Wiesenfeld, Physica D 48, 102 (1991)
M. Bennett, K. Wiesenfeld, Physica D 192, 196 (2004)
K. Wiesenfeld, J. Swift, Phys. Rev. E 51, 1020 (1995)
S. Watanabe, S. Strogatz, Physica D 74, 197 (1994)
J. Swift, S. Strogatz, K. Wiesenfeld, Physica D 55, 239 (1992)
S. Watanabe, J. Swift, J. Nonlinear Sci. 7, 503 (1997)
A. Barone, G. Paterno, Physics and Applications of the Josephson Effect (J. Wiley, New York, 1982)
A. Bulsara, J. Acebron, W. Rappel, A. Hibbs, L. Kunstmanas, M. Krupka, Physica A 325, 220 (2003)
J.L. Aven, Networks of coupled SQUID magnetometers, Master’s thesis, San Diego State University, 2006
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Berggren, S., Palacios, A. Analytical approximations to the dynamics of an array of coupled DC SQUIDs. Eur. Phys. J. B 87, 83 (2014). https://doi.org/10.1140/epjb/e2014-50065-9
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DOI: https://doi.org/10.1140/epjb/e2014-50065-9