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On Superoscillations Longevity: A Windowed Fourier Transform Approach

  • Conference paper
Quantum Theory: A Two-Time Success Story

Abstract

In this paper we prove that the evolution of a superoscillating sequence of functions Y N (x), which we take as the initial value of the Schrödinger equation, remains superoscillating for all values of the time \(t<\sqrt{N}\). We prove this by using a windowed Fourier transform approach. [Editor’s note: for a video of the talks given by Prof. Aharonov at the Aharonov-80 conference in 2012 at Chapman University, see quantum.chapman.edu/talk-3 and quantum.chapman.edu/talk-30. For the subject of super-oscillations, the editors also recommend the video of the talk given by Prof. Berry at the Aharonov-80 conference in 2012 at Chapman University, see quantum.chapman.edu/talk-6 along with the talk by Prof. Casher, see quantum.chapman.edu/talk-21.]

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References

  1. M.V. Berry, Faster than Fourier, in Quantum Coherence and Reality, ed. by J.S. Anandan, J.L. Safko (World Scientific, Singapore, 1994), pp. 55–65 (in celebration of the 60th Birthday of Yakir Aharonov)

    Google Scholar 

  2. Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen, Some mathematical properties of superoscillations. J. Phys. A 44, 365304 (2011)

    Article  MathSciNet  Google Scholar 

  3. Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen, The mathematics of superoscillations, in preparation

    Google Scholar 

  4. M.V. Berry, J. Phys. A 27, L391–L398 (1994)

    Article  ADS  MATH  Google Scholar 

  5. M.V. Berry, J. Phys. A 43, 415302 (2010)

    Article  MathSciNet  Google Scholar 

  6. M.V. Berry, N. Brunner, S. Popescu, P. Shukla, J. Phys. A 44, 492001 (2011)

    Article  MathSciNet  Google Scholar 

  7. M.V. Berry, J. Phys. A 45, 185308 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  8. Y. Aharonov, J. Anandan, S. Popescu, L. Vaidman, Phys. Rev. Lett. 65, 2965 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  9. P.J.S.G. Ferreira, A. Kempf, J. Phys. A 37, 12067–12076 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. P.J.S.G. Ferreira, A. Kempf, IEEE Trans. Signal Process. 54, 3732–3740 (2006)

    Article  ADS  Google Scholar 

  11. J. Lindberg, Mathematical concepts of optical superresolution. J. Opt. 14, 083001 (2012)

    Article  ADS  Google Scholar 

  12. W. Qiao, J. Phys. A 29, 2257–2258 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. M.V. Berry, J. Opt. 15(4), 044006 (2013)

    Article  ADS  Google Scholar 

  14. M.V. Berry, J. Phys. A: Math. Theor. 46(20), 205203 (2013)

    Article  ADS  Google Scholar 

  15. M.V. Berry, S. Popescu, J. Phys. A 39, 6965–6977 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen, On the Cauchy problem for the Schrödinger equation with superoscillatory initial data. J. Math. Pures Appl. 99, 165–173 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Y. Aharonov, F. Colombo, S. Nussinov, I. Sabadini, D.C. Struppa, J. Tollaksen, Superoscillations phenomenona in SO(3). Proc. R. Soc. A, Math. Phys. Eng. Sci. 468, 3587–3600 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  18. Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen, On some operators associated to superoscillations. Complex Anal. Oper. Theory 7, 1299–1310 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. M.R. Dennis, C. Hamilton, Opt. Lett. 33, 2976–2978 (2008)

    Article  ADS  Google Scholar 

  20. E.T.F. Rogers, J. Lindberg, T. Roy, S. Savo, J.E. Chad, M.R. Dennis, N.I. Zheludev, Nat. Mater. 11, 432–435 (2012)

    Article  ADS  Google Scholar 

  21. Y. Aharonov, S. Popescu, J. Tollaksen, A time-symmetric formulation of quantum mechanics. Phys. Today 63(11), 27–32 (2010)

    Article  Google Scholar 

  22. Y. Aharonov, D.Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351–1354 (1988)

    Article  ADS  Google Scholar 

  23. M. Büttiker, Phys. Rev. B 27, 6178 (1983)

    Article  ADS  Google Scholar 

  24. Y. Aharonov, S. Popescu, D. Rohrlich. Preprint, TAUP 184790, Tel Aviv University (1990)

    Google Scholar 

  25. Y. Aharonov, L. Davidovich, N. Zagury, Phys. Rev. A 48, 1687 (1993)

    Article  ADS  Google Scholar 

  26. Y. Aharonov, P.G. Bergmann, J.L. Lebowitz, Phys. Rev. B 134, 1410 (1964)

    Article  MathSciNet  ADS  Google Scholar 

  27. Y. Aharonov, L. Vaidman, Phys. Rev. A 41, 11–20 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  28. Y. Aharonov, L. Vaidman, Time in Quantum Mechanics, ed. by J. Muga, R. Sala Mayato, I. Egusquiza. quant-ph/0105101

  29. N. Ritchie, J. Story, R. Hulet, Phys. Rev. Lett. 66, 1107 (1991)

    Article  ADS  Google Scholar 

  30. A. Parks, D. Cullen, D. Stoudt, Proc. R. Soc. Lond. A 454, 2997 (1998)

    Article  ADS  MATH  Google Scholar 

  31. K. Resch, J. Lundeen, A. Steinberg, Phys. Lett. A 324, 125 (2004)

    Article  ADS  MATH  Google Scholar 

  32. Q. Wang, F. Sun, Y. Zhang, J. Li, Y. Huang, G. Guo, Phys. Rev. A 73, 023814 (2006)

    Article  ADS  Google Scholar 

  33. O. Hosten, P. Kwiat, Science 319, 787 (2008)

    Article  ADS  Google Scholar 

  34. K. Yokota, T. Yamamoto, M. Koashi, N. Imoto, New J. Phys. 11, 033011 (2009)

    Article  ADS  Google Scholar 

  35. P. Dixon, D. Starling, A. Jordan, J. Howell, Phys. Rev. Lett. 102, 173601 (2009)

    Article  ADS  Google Scholar 

  36. J. Tollaksen, in Quantum Information and Computation V, ed. by E. Donkor, A. Pirich, H. Brandt. Proc. of SPIE, vol. 6573 (2007) (CID 6573-33)

    Google Scholar 

  37. M.V. Berry, M.R. Dennis, J. Phys. A 42, 022003 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  38. M.V. Berry, P. Shukla, Pragya, J. Phys. A 45, 015301 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  39. M.V. Berry, M.R. Dennis, B. McRoberts, P. Shukla, J. Phys. A 44, 205301 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  40. Y. Aharonov, A. Botero, Quantum averages of weak values. Phys. Rev. A 72(5), 052111 (2005)

    Article  ADS  Google Scholar 

  41. A. Botero, Sampling weak values: a non-linear Bayesian model for non-ideal quantum measurements. arXiv:quant-ph/0306082

  42. M.V. Berry, P. Shukla, J. Phys. A 43, 354024 (2010)

    Article  MathSciNet  Google Scholar 

  43. Y. Aharonov, S. Massar, S. Popescu, J. Tollaksen, L. Vaidman, Phys. Rev. Lett. 77, 983 (1996)

    Article  ADS  Google Scholar 

  44. Y. Aharonov, A. Botero, S. Popescu, B. Reznik, J. Tollaksen, Phys. Lett. A 301, 130–138 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. S.N. Nussinov, J. Tollaksen, Phys. Rev. D 78, 036007 (2008)

    Article  ADS  Google Scholar 

  46. J. Tollaksen, Robust weak measurements on finite samples. J. Phys. Conf. Ser. 70, 012015 (2007)

    Article  ADS  Google Scholar 

  47. J. Tollaksen, J. Phys. A 40, 9033–9066 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen, Superoscillating sequences as solutions of generalized Schrödinger equations. Preprint (2013)

    Google Scholar 

  49. D. Gabor, Theory of communication. J. Inst. Electr. Eng., Lond. 93, 429–457 (1946)

    Google Scholar 

  50. S. Mallat, A Wavelet Tour of Signal Processing (Academic Press, San Diego, 1998)

    MATH  Google Scholar 

  51. Y. Aharonov, D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed (Wiley-VCH, Weinheim, 2005)

    Book  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute for Quantum Studies and Schmid College of Science and Technology, Chapman University, Orange, 92866, CA, USA

    Y. Aharonov & J. Tollaksen

  2. Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9 20133, Milano, Italy

    F. Colombo & I. Sabadini

  3. Institute for Quantum Studies and Schmid College of Science, Chapman University, Orange, 92866, CA, USA

    D. C. Struppa

Authors
  1. Y. Aharonov
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  2. F. Colombo
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  3. I. Sabadini
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  4. D. C. Struppa
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  5. J. Tollaksen
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Corresponding author

Correspondence to D. C. Struppa .

Editor information

Editors and Affiliations

  1. Schmid College of Science and Technology, Chapman University, Orange, USA

    Daniele C. Struppa

  2. Schmid College of Science and Technology, Chapman University, Orange, USA

    Jeffrey M. Tollaksen

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Aharonov, Y., Colombo, F., Sabadini, I., Struppa, D.C., Tollaksen, J. (2014). On Superoscillations Longevity: A Windowed Fourier Transform Approach. In: Struppa, D., Tollaksen, J. (eds) Quantum Theory: A Two-Time Success Story. Springer, Milano. https://doi.org/10.1007/978-88-470-5217-8_20

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