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Circuit Approach for Simulation of EM-quantum Components

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Applications of Advanced Electromagnetics

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 169))

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Abstract

This Chapter is on the circuit approach to describe the quantummechanical phenomena. Being proposed by G. Kron many years ago, this technique is now a very powerful tool for modeling and design of hybrid electronics integrating the classical and quantum-mechanical components. The linear and non-linear Schrödinger equations are transformed into the first-order partial differential equations with respect to currents and voltages, and the obtained equivalent circuits are modeled using a commercially available simulator. The approach is pertinent for seamless simulation of the future-generation integration, although the main attention in this Chapter is paid to the modeling of trapped Bose-Einstein condensates. References -108. Figures -34. Pages -54.

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References

  1. Tour, J.M.: Molecular Electronics. World Sci. (2003)

    Google Scholar 

  2. Deleonibus, S. (ed.): Electronic Device Architectures for the Nano-CMOS Era. World Sci. (2008)

    Google Scholar 

  3. Benci, V., Fortunato, D.: An eigenvalue problem for the Schrödinger-Maxwell equations. Topological Methods in Nonlinear Analysis 11, 283–293 (1998)

    MathSciNet  MATH  Google Scholar 

  4. Ginbre, B., Velo, G.: Long range scattering for the Maxwell-Schrödinger system with large magnetic field data and small Schrödinger data. Publ. RIMS, Kyoto Univ. 42, 421–459 (2006)

    Article  Google Scholar 

  5. Yang, J., Sui, W.: Solving Maxwell-Schrödinger equations for analyses of nano-scale devices. In: Proc. 37th Europ. Microw. Conf., pp. 154–157 (2007)

    Google Scholar 

  6. Pieratoni, B., Mencarelli, D., Rozzi, T.: A new 3-D transmission line matrix scheme for the combined Schrödinger-Maxwell problem in the electronic/electromagnetic characterization of nanodevices. IEEE Trans., Microwave Theory Tech. 56, 654–662 (2008)

    Article  Google Scholar 

  7. Pieratoni, B., Mencarelli, D., Rozzi, T.: Boundary immitance operators for the Schrödinger-Maxwell problem of carrier dynamics in nanodevices. IEEE Trans., Microwave Theory Tech. 57, 1147–1155 (2009)

    Article  Google Scholar 

  8. Mastorakis, N.E.: Solution of the Schrödinger-Maxwell equations via finite elements and genetic algorithms with Nelder-Mead. WSEAS Trans. Math. 8, 169–176 (2009)

    MathSciNet  Google Scholar 

  9. Kron, G.: Electric circuit model of the Schrödinger equation. Phys. Rev. (1&2) (1945)

    Google Scholar 

  10. Sanada, H., Suzuki, M., Nagai, N.: Analysis of resonant tunneling using the equivalent transmission-line model. IEEE J. Q. Electron. 33, 731–741 (1977)

    Article  Google Scholar 

  11. Anwar, A.F.M., Khondker, A.N., et al.: Calculation of the transversal time in resonant tunneling devices. J. Appl. Phys. 65, 2761–2765 (1989)

    Article  Google Scholar 

  12. Kaji, R., Koshiba, M.: Equivalent network approach for guided electron waves in quantum-well structures and its application to electron-wave directional couplers. IEEE J. Quant. Electron 31, 1036–1043 (1994)

    Article  Google Scholar 

  13. Civalleri, P.P., Gilli, M., Bonnin, M.: Equivalent circuits for two-state quantum systems. Int. J. Circ. Theory Appl. 35, 265–280 (2007)

    Article  Google Scholar 

  14. Kouzaev, G.A.: Hertz vectors and the electromagnetic-quantum equations. Mod. Phys. Lett. B 24(24), 2117–2212 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kouzaev, G.A.: Calculation of linear and non-linear Schrödinger equations by the equivalent network approach and envelope technique. Modern Phys. Lett. B 24, 29–38 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Matyas, A., Jirauschek, C., Perretti, P., et al.: Linear circuit models for on-chip quantum electrodynamics. IEEE Trans., Microw. Theory Tech. 59, 65–71 (2011)

    Article  Google Scholar 

  17. Kouzaev, G.A., Nazarov, I.V., Kalita, A.V.: Unconventional logic elements on the base of topologically modulated signals. El. Archive, http://xxx.arXiv.org/abs/physics/9911065

  18. Kouzaev, G.A., Lebedeva, T.A.: Multivalued and quantum logic modeling by mode physics and topologically modulated signals. In: Proc. Int. Conf. Modelling and Simulation, Las Palmas de Grand Canaria, Spain, September 25-27 (2000), http://www.dma.ulpgc.es/ms2000

  19. Kouzaev, G.A.: Predicate and pseudoquantum gates for amplitude-spatially modulated electromagnetic signals. In: Proc. 2001 IEEE Int. Symp. Intelligent Signal Processing and Commun. Systems, Nashville, Tennessee, USA, November 20-23 (2001)

    Google Scholar 

  20. Kouzaev, G.A.: Qubit logic modeling by electronic gates and electromagnetic signals. El. Archive (2001), http://xxx.arXiv.org/abs/quant-ph/0108012

  21. Advanced Design System 2008. Agilent Corp. (2008)

    Google Scholar 

  22. A User Guide to Envelope Following Analysis Using Spectre RF. Cadence Corp. (2007)

    Google Scholar 

  23. Visscher, P.B.: A fast explicit algorithm for the time-dependent Schrödinger equation. Comp. Phys. 5/6, 596–598 (1991)

    Google Scholar 

  24. Frank, T.D.: Nonlinear Fokker-Planck Equations: Fundamentals and Applications. Springer, Berlin (2005)

    Google Scholar 

  25. Norbe, F.D., Rego-Monteiro, M.A., Tsallis, C.: A generalized nonlinear Schroedinger equation: Classical field-theoretic approach. Eur. Phys. Lett. 97(1-5), 41001 (2012)

    Google Scholar 

  26. Belevitch, V.: Classical Network Theory. Holden-Day (1968)

    Google Scholar 

  27. Galizkyi, V.M., Kornakov, B.M., Kogan, V.I.: Tasks to Solve in Quantum Mechanics (Zadachi po Kvantovoy Mekhanike), Nauka (1981) (in Russian)

    Google Scholar 

  28. Gross, E.P.: Structure of a quantized vortex in boson systems II. Nouvo Cimento 20, 454–457 (1961)

    Article  MATH  Google Scholar 

  29. Pitaevskii, L.V.: Vortex lines in an imperfect Bose gas. Soviet Phys. JETP 13, 451–454 (1961)

    MathSciNet  Google Scholar 

  30. Pitaevskii, L.P., Stringari, S.: Bose-Einstein Condensation. Clareton Press (2003)

    Google Scholar 

  31. Ueda, M.: Fundamentals and New Frontiers of Bose-Einstein Condensation. World Scientific (2010)

    Google Scholar 

  32. Vengalattore, M., Higbie, J.M., Leslie, S.R., et al.: High-Resolution Magnetometry with a spinor Bose-Einstein Condensate. Phys. Rev. Lett. 98, 200801 (2007)

    Article  Google Scholar 

  33. Simmonds, R.W., Marchenkov, A., Hoskinson, E., et al.: Quantum interference of super fluid 3He. Nature 412, 55–58 (2001)

    Article  Google Scholar 

  34. Seaman, T., Krämer, M., Anderson, D.Z., et al.: Atomtronics: ultracold-atom analogs of electronic devices. Phys. Rev. A 75, 023615 (2007)

    Article  Google Scholar 

  35. Stickney, J.A., Anderson, D.Z., Zozulya, A.A.: Transistorlike behavior of a Bose-Einstein condensate in a triple-well potential. Phys. Rev. A 75, 013608 (2007)

    Article  Google Scholar 

  36. Ramanathan, A., Wright, K.C., Muniz, S.R., et al.: Superflow in a toroidal Bose-Einstein condensate: an atom circuit with a tunable weak link. Phys. Rev. Lett. 106, 13041 (2001)

    Google Scholar 

  37. Farkas, M., Hudek, K.M., Salim, E.A., et al.: A compact, transportable, microchip-based system for high repetition rate production of Bose-Einstein condensates. App. Phys. Lett. 96, 093102 (2001)

    Article  Google Scholar 

  38. Cataliotti, F., Burger, S., Fort, C., et al.: Josephson junction arrays with Bose-Einstein condensates. Science 293, 843–846 (2001)

    Article  Google Scholar 

  39. Succi, S., Toschi, F., Tosi, M.P., et al.: Bose-Einstein condensates and the numerical solution of Gross-Pitaevskii equation. IEEE Comput. Sci. Eng. 7, 48–57 (2005)

    Google Scholar 

  40. Cerimele, M.M., Chiofalo, M.L., Pistella, F., et al.: Numerical solution of the Gross-Pitaevskii equation using an explicit finite-difference scheme: an application to trapped Bose-Einstein condensates. Phys. Rev. E 62, 1382–1389 (2000)

    Article  Google Scholar 

  41. Bao, W., Tang, W.: Ground-state solution of Bose-Einstein condensate by directly minimizing the energy functional. J. Comput. Phys. 187, 230–254 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  42. Kron, G.: Numerical solution of ordinary and partial differential equations by means of equivalent circuits. J. Appl. Phys. 16, 172–186 (1945)

    Article  MathSciNet  MATH  Google Scholar 

  43. Kron, G.: Equivalent circuit of the field equations of Maxwell. In: Proc. I. R. E., pp. 289–299 (1944)

    Google Scholar 

  44. Dragoman, D., Dragoman, M.: Quantum-classical Analogies. Springer (2004)

    Google Scholar 

  45. Kouzaev, G.A.: Co-design of quantum and electronic integrations by available circuit simulators. In: Proc. 13th Int. Conf. Circuits, Rodos, Greece, pp. 152–156 (2009)

    Google Scholar 

  46. Holland, M.J., Jin, D.S., Chiofalo, M.L., et al.: Emergence of interaction effects in Bose-Einstein condensation. Phys. Rev. Lett. 78, 3801–3805 (1997)

    Article  Google Scholar 

  47. Bogolubov, N.: J. Phys 11, 23 (1947) (in Russian)

    MathSciNet  Google Scholar 

  48. Pethick, C.J., Smith, H.: Bose-Einstein Condensation in Dilute Gases. Cambridge Press (2003)

    Google Scholar 

  49. Chevy, F., Dalibard, J.: Rotating Bose-Einstein condensates. Europhysicsnews 37, 12–16 (2006)

    Google Scholar 

  50. Rozanov, N.N., Rozhdestvenkyi, Y.V., Smirnov, V.A., et al.: Atomic “Needles” and “Bullets” of the Bose-Einstein condensate and forming of nano-size structures. Pisma v ZHETF- Lett. J. Exper. Theor. Phys. 77, 89–92 (2003) (in Russian)

    Google Scholar 

  51. Anderson, M.H., Ensher, J.R., Matthews, M.R., Wieman, C.E., Cornell, E.A.: Observation of Bose–Einstein condensation in a dilute atomic vapor. Science 269(5221), 198–201 (1995)

    Article  Google Scholar 

  52. Davis, K.B., Mewes, M.-O., Andrews, M.R., van Druten, N.J., Durfee, D.S., Kurn, D.M., Ketterle, W.: Bose–Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75, 3969–3973 (1995)

    Article  Google Scholar 

  53. Balykin, V.I., Minogin, V.G., Letokhov, V.S.: Electromagnetic trapping of cold atoms. Rep. Prog. Phys. 61, 1429–1510 (2000)

    Article  Google Scholar 

  54. Chu, S.: Laser manipulations of atoms and particles. Science 253, 861–866 (1991)

    Article  Google Scholar 

  55. Cohen-Tannoudji, C., Guerry-Odelin, D.: Advances in Atomic Physics: an Overview. World Scientific (2011)

    Google Scholar 

  56. Phillips, W.D.: Nobel lecture: Laser cooling and trapping of neutral atoms. Rev. Mod. Phys. 70, 721–741 (1998)

    Article  Google Scholar 

  57. Friedman, N., Kaplan, A., Davidson, N.: Dark optical traps for cold atoms. Adv. Atomic, Molec., Opt. Phys. 48, 99–151 (2002)

    Article  Google Scholar 

  58. Noh, H.-R., Jhe, W.: Atom optics with hollow optical systems. Phys. Reports 372, 269–317 (2002)

    Article  Google Scholar 

  59. Kuhr, S., Alt, W., Schrader, D., et al.: Deterministic delivery of a single atom. Science 293, 278–280

    Google Scholar 

  60. Mandel, A., Greiner, M., Widera, A., et al.: Controlled collisions for multi-particle entanglement of optically trapped atoms. Nature 425, 937–940 (2003)

    Article  Google Scholar 

  61. Schrader, D., Dotsenko, I., Khudaverdyan, M., et al.: Neutral atom quantum register. Phys. Rev. Lett. 93(1-4), 150501

    Google Scholar 

  62. Bloch, I.: Exploring quantum matter with ultracold atoms in optical lattices. J. Phys. B 38, S629–S643 (2005)

    Article  Google Scholar 

  63. Bloch, I., Dalibard, J., Zwerger, W.: Many-body physics with ultra-cold gases. Rev. Mod. Phys., 885–964 (2008)

    Google Scholar 

  64. Bergman, T., Erez, G., Metcalf, H.J.: Magnetostatic trapping fields for neutral atoms. Phys. Rev. A 35, 1535–1546 (1987)

    Article  Google Scholar 

  65. AMPERES Program Guide. Integrated Software Eng. Inc. (2006)

    Google Scholar 

  66. Sand, K.J.: On the design and simulation of electromagnetic traps and guides for ultra-cold matter. PhD Thesis, NTNU, Trondheim, Norway, 252 p (2010)

    Google Scholar 

  67. Kouzaev, G.A., Sand, K.J.: RF controllable Ioffe-Pritchard trap for cold dressed atoms. Modern Phys. Lett. B 21, 59–68 (2007)

    Article  MATH  Google Scholar 

  68. Thomas, N.R., Foot, C.J., Wilson, A.C.: Double-well magnetic trap for Bose-Einstein condensates. ArXiv: cond-mat/01108169 (2001)

    Google Scholar 

  69. Tiecke, T.G., Kemmann, M., Buggle, C., et al.: Bose-Einstein condensation in a magnetic double-well potential. ArXiv: cond-mat/0211604 (2002)

    Google Scholar 

  70. Rechel, J., Hansel, W., Hommelholf, P., et al.: Applications of integrated magnetic microtraps. Appl. Phys. B 72, 81–89 (2001)

    Article  Google Scholar 

  71. Jones, M.P.A., Vale, C.J., Sahagun, D., et al.: Cold atoms probe the magnetic field near a wire. J. Phys. B 37, L15–L20 (2004)

    Article  Google Scholar 

  72. Crookston, M.B., Baker, P.M., Robinson, M.P.: A microstrip ring trap for cold atoms. J. Phys. B 38, 3227–3289 (2005)

    Article  Google Scholar 

  73. Koukharenko, E., Mktadir, Z., Kraft, M., et al.: Microfabrication of gold wires for atom guides. Sensors and Actuators A 115, 600–607

    Google Scholar 

  74. Henkel, Wilkens, M.: Heating of trapped atoms near thermal surfaces. Europhys. Lett. 47, 414–420 (1999)

    Article  Google Scholar 

  75. Fermani, R., Scheel, S., Knight, P.L.: Trapping cold atoms near carbon nanotubes: Thermal spin flips and Casimir-Polder potential. Phys. Rev. A 75(1-7), 062905 (2007)

    Article  Google Scholar 

  76. Bostroem, M., Sernelius, B.E., Brevik, I., et al.: Retardation turns the van der Waals attraction into a Casimir repulsion as close as 3 nm. Phys. Rev. A 85(1-4), 010701

    Google Scholar 

  77. Kouzaev, G.A., Sand, K.J.: 3D multicell designs for registering of Bose-Einstein condensate clouds. Modern Phys. Lett. 22(25), 2469–2479 (2008)

    Article  Google Scholar 

  78. Shi, Y.: Entanglement between Bose-Einstein condensates. Int. J. Modern. Phys. B 15, 3007–3030 (2001)

    Article  Google Scholar 

  79. Yalabik, M.C.: Nonlinear Schrödinger equation for quantum computation. Modern Physics Lett. B 20, 1099–1106 (2006)

    Article  MATH  Google Scholar 

  80. Albiez, M., Gati, R., Foeling, J., et al.: Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction. Phys. Rev. Lett. 95(1-4), 010402 (2005)

    Article  Google Scholar 

  81. Levy, S., Lahoud, E., Shomroni, I., et al.: The a.c. and d.c. Josephson effects in a Bose-Einstein condensate. Nature 449, 579–583 (2007)

    Article  Google Scholar 

  82. Muskat, E.: Dressed neutrons. Phys. Rev. Lett. 58, 2047–2050 (1987)

    Article  Google Scholar 

  83. Zobay, O., Garraway, B.M.: Two-dimensional atom trapping in field-induced adiabatic potentials. Phys. Rev. Lett. 86, 1195–1198 (2001)

    Article  Google Scholar 

  84. Colombe, Y., Knyazchyan, E., Morizot, O., et al.: Ultracold atoms confined in rf-induced two-dimensional trapping potentials. arXiv:quant-ph/0403006

    Google Scholar 

  85. Courteille, P.W., Deh, B., Fortag, J., et al.: Highly versatile atomic micro traps generated by multifrequency magnetic field modulation. arXiv:quant-ph/0512061

    Google Scholar 

  86. Schumm, T., Hofferberth, S., Andersson, L.M., et al.: Matter-wave interferometry in a double well on an atom chip. Nature Physics 1, 57–62 (2005)

    Article  Google Scholar 

  87. Lesanovsky, I., Schumm, T., Hofferberth, S., et al.: Adiabatic radio frequency potentials for coherent manipulation of matter waves. ArXiv:physics/0510076

    Google Scholar 

  88. Ol’shanii, M.A., Ovchinnikov, Y.V., Letokhov, V.S.: Laser guiding of atoms in a hollow optical fiber. Opt. Commun. 98, 77–79 (1993)

    Article  Google Scholar 

  89. Renn, M.J., Mongomery, D., Vdovin, O., et al.: Laser-Guided atoms in hollow-core optical fibers. Phys. Rev. Lett. 75, 3253–3256 (1995)

    Article  Google Scholar 

  90. Renn, M.J., Donley, E.A., Cornell, E.A., et al.: Evanescent-wave guiding of atoms in hollow optical fibers. Phys. Rev. A 53, R648–R651 (1996)

    Article  Google Scholar 

  91. Song, Y., Milam, D., Hill III, W.T.: Long, narrow all-light atom guide. Opt. Lett. 24, 1805–1807 (1999)

    Article  Google Scholar 

  92. Myatt, C.J., Newbury, N.R., Ghrist, R.W., et al.: Multiply loaded magneto-optical trap. Opt. Lett. 21, 290–292

    Google Scholar 

  93. Goepfert, A., Lison, F., Schutze, R., et al.: Efficient magnetic guiding and deflection of atomic beams with moderate velocities. Appl. Phys. B 69, 217–222

    Google Scholar 

  94. Key, M., Hughes, I.G., Rooijakkers, W., et al.: Propagation of cold atoms along a miniature magnetic guide. Phys. Rev. Lett. 84, 1371–1373 (2000)

    Article  Google Scholar 

  95. Teo, B.K., Raithel, G.: Loading mechanism for atomic guides. Phys. Rev. A 63(1-4), 031402 (2001)

    Article  Google Scholar 

  96. Yung-Kuo, L.: Problems and Solutions on Electromagnetism. World Scientific (1993)

    Google Scholar 

  97. Subbotin, M.V., Balykin, V.I., Laryushin, D.L., et al.: Laser controlled atom waveguide as a source of ultracold atoms. Opt. Commun. 139, 107 (1997)

    Article  Google Scholar 

  98. Greiner, M., Bloch, I., Haensh, T.W., et al.: Magnetic transport of trapped cold atoms over a large distance. Phys. Rev. A 63(1-4), 0131401 (2001)

    Google Scholar 

  99. Kouzaev, G.A., Sand, K.J.: Inter-wire transfer of cold dressed atoms. Modern Phys. Lett. B 21, 1653–1665 (2007)

    Article  Google Scholar 

  100. Weinstein, J.D., Librecht, K.G.: Microscopic magnetic traps for neutral particles. Phys. Rev. A 52, 4004–4009 (1995)

    Article  Google Scholar 

  101. Thywissen, J.J., Olshanii, M., Zabow, G., et al.: Microfabricated magnetic waveguides for neutral atoms. Eur. Phys. J. D 7, 361–367 (1999)

    Article  Google Scholar 

  102. Allwood, D.A., Schrefl, T., Hrkac, G., et al.: Mobile atom traps using nanowires. Appl. Phys. Lett. 89(1-3), 014102 (2006)

    Article  Google Scholar 

  103. Dekker, N.H., Lee, C.S., Lorent, V., et al.: Guiding neutral atoms on a chip, vol. 84, pp. 1124–1127 (2000)

    Google Scholar 

  104. Tonyshkin, A., Prentiss, M.: Straight macroscopic magnetic guide for cold atom interferometer. J. Appl. Phys. 108(1-5), 094904 (2010)

    Article  Google Scholar 

  105. Bongs, K., Burger, S., Dettmer, S., et al.: Waveguide for Bose-Einstein condensates. Phys. Rev. A 63(1-4), 031602 (2001)

    Article  Google Scholar 

  106. Treutlein, P., Hommelhoff, P., Steinmetz, T., et al.: Coherence in microstrip traps. Phys. Rev. Lett. 92(1-4), 203005 (2004)

    Article  Google Scholar 

  107. Treutlein, P., Steinmetz, T., Colombe, Y., et al.: Quantum information processing in optical lattices and magnetic microtraps. Fortschr. Phys. 54, 702–718 (2006)

    Article  Google Scholar 

  108. Boehi, P., Riedel, M.F., Hoffrogge, J., et al.: Coherent manipulation of Bose-Einstein condensates with state-dependent microwave potentials on an atom chip. Nature Physics 5, 592–597 (2009)

    Article  Google Scholar 

  109. Sun, Y., Tan, W., Jiang, H.-T., et al.: Metamaterial analog of quantum interference: From electromagnetically induced transparency to absorbtion. EPLA 98, 6407 (1-6) (2012)

    Google Scholar 

  110. Rangelow, A.A., Suchowski, H., Silberberg, Y., et al.: Wireless adiabatic power transfer. Annals of Phys. 326, 626–633 (2011)

    Article  Google Scholar 

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  1. Department of Electronics and Telecommunications , Norwegian University of Science and Technology, O.S. Bragstads plass 2B, 7491, Trondheim, Norway

    Guennadi A. Kouzaev

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Kouzaev, G.A. (2013). Circuit Approach for Simulation of EM-quantum Components. In: Applications of Advanced Electromagnetics. Lecture Notes in Electrical Engineering, vol 169. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30310-4_9

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