The European Physical Journal Special Topics

, Volume 226, Issue 7, pp 1593–1601 | Cite as

Quantum revival for elastic waves in thin plate

Open Access
Regular Article
Part of the following topical collections:
  1. From Ill-condensed Matter to Mesoscopic Wave Propagation


Quantum revival is described as the time-periodic reconstruction of a wave packet initially localized in space and time. This effect is expected in finite-size systems which exhibit commensurable discrete spectrum such as the infinite quantum well. Here, we report on the experimental observation of full and fractional quantum revival for classical waves in a two dimensional cavity. We consider flexural waves propagating in thin plates, as their quadratic dispersion at low frequencies mimics the dispersion relation of quantum systems governed by Schrödinger equation. Time-dependent excitation and measurement are performed at ultrasonic frequencies and reveal a periodic reconstruction of the initial elastic wave packet.

Supplementary material


  1. 1.
    R. Robinett, Quantum wave packet revivals, Phys. Rep. 392, 1 (2004)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    E. Jaynes, F. Cummings, Comparison of quantum, semiclassical radiation theories with application to the beam maser, Proc. IEEE 51, 89 (1963)CrossRefGoogle Scholar
  3. 3.
    J.H. Eberly, N.B. Narozhny, J.J. Sanchez-Mondragon, Periodic Spontaneous Collapse, Revival in a Simple Quantum Model, Phys. Rev. Lett. 44, 1323 (1980)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    N. Narozhny, J. Sanchez-Mondragon, J. Eberly, Coherence versus incoherence: Collapse revival in a simple quantum model, Phys. Rev. A 23, 236 (1981)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Parker, C. Stroud, Coherence decay of Rydberg wave packets, Phys. Rev. Lett. 56, 1 (1986)CrossRefGoogle Scholar
  6. 6.
    I. Averbukh, N. Perelman, Fractional revivals: Universality in the long-term evolution of quantum wave packets beyond the correspondence principle dynamics, Phys. Lett. A 139, 449 (1989)ADSCrossRefGoogle Scholar
  7. 7.
    S. Tomsovic, J.H. Lefebvre, Can wave packet revivals occur in chaotic quantum systems?, Phys. Rev. Lett. 79, 3629 (1997)ADSCrossRefGoogle Scholar
  8. 8.
    G. Rempe, H. Walther, N. Klein, Observation of quantum collapse and revival in a one-atom maser, Phys. Rev. Lett. 58, 353 (1987)ADSCrossRefGoogle Scholar
  9. 9.
    J. Yeazell, M. Mallalieu, C. Stroud, Observation of the collapse and revival of a Rydberg electronic wave packet, Phys. Rev. Lett. 64, 2007 (1990)ADSCrossRefGoogle Scholar
  10. 10.
    J. Wals, H. Fielding, J. Christian, L. Snoek, W. van der Zande, H. van Linden van den Heuvell, Observation of Rydberg wave packet dynamics in a Coulombic magnetic field, Phys. Rev. Lett. 72, 3783 (1994)ADSCrossRefGoogle Scholar
  11. 11.
    V. Krueckl, T. Kramer, Revivals of quantum wave packets in graphene, J. Phys. 11, 093010 (2009)Google Scholar
  12. 12.
    M. Vrakking, D. Villeneuve, A. Stolow, Observation of fractional revivals of a molecular wave packet, Phys. Rev. A 54, R37 (1996)ADSCrossRefGoogle Scholar
  13. 13.
    G. Kirchmair, B. Vlastakis, Z. Leghtas, S.E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S.M. Girvin, R.J. Schoelkopf, Observation of quantum state collapse revival due to the single-photon Kerr effect, Nature 495, 205 (2013)ADSCrossRefMATHGoogle Scholar
  14. 14.
    H. Talbot, Facts relating to optical science, Phil. Mag. 9, 401407 (1836)Google Scholar
  15. 15.
    J. Winthrop, C. Worthington, Theory of Fresnel images. I. Plane periodic objects in monochromatic light, JOSA 55, (1965)Google Scholar
  16. 16.
    M.V. Berry, S. Klein, Integer fractional fractal Talbot effects, J. Mod. Opt. 43, 2139 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    N.K. Efremidis, D.N. Christodoulides, Revivals in engineered waveguide arrays, Opt. Commun. 246, 345 (2005)ADSCrossRefGoogle Scholar
  18. 18.
    G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duò, S. Longhi, Experimental Observation of a Photon Bouncing Ball, Phys. Rev. Lett. 102, 180402 (2009)ADSCrossRefGoogle Scholar
  19. 19.
    D.L. Aronstein, C.R. Stroud, Fractional wave-function revivals in the infinite square well, Phys. Rev. A 55, 4526 (1997)ADSCrossRefGoogle Scholar
  20. 20.
    D. Royer, E. Dieulesaint, Ondes Élastiques Dans Les Solides Propagation Libre et Guidée (Elsevier Masson, 1996)Google Scholar
  21. 21.
    A.W. Leissa, Vibration of Plates (NASA, 1969), SP160Google Scholar
  22. 22.
    A. Venugopalan, G. Agarwal, Superrevivals in the quantum dynamics of a particle confined in a finite square-well potential, Phys. Rev. A 59, 1413 (1999)ADSCrossRefGoogle Scholar
  23. 23.
    J. Virieux, P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method, Geophys. 51, 4 (1986)CrossRefGoogle Scholar
  24. 24.
    E. Bossy, M. Talmant, P. Laugier, Three-dimensional simulations of ultrasonic axial transmission velocity measurement on cortical bone models, J. Acoust. Soc. Am. 115, 2314 (2004)ADSCrossRefGoogle Scholar
  25. 25.
    See Supplementary Material for movies of the vertical displacement for point source located on the diagonal at the center of the plateGoogle Scholar
  26. 26.
    M.A. Doncheski, S. Heppelmann, R.W. Robinett, D.C. Tussey, Wave packet construction in two-dimensional quantum billiards: Blueprints for the square equilateral triangle circular cases, Am. J. Phys. 71, 541 (2003)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    F. Anselmet, P.O. Mattei, Acoustique Aaroacoustique et Vibrations (ISTE Ltd. – Wiley & Sons, 2015)Google Scholar

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© The Author(s) 2017

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Marc Dubois
    • 1
  • Gautier Lefebvre
    • 1
  • Patrick Sebbah
    • 1
    • 2
  1. 1.Institut Langevin, ESPCI ParisTech CNRS UMR7587, 1 rue JussieuParis cedex 05France
  2. 2.Department of PhysicsThe Jack and Pearl Resnick Institute for Advanced Technology, Bar-Ilan UniversityRamat-GanIsrael

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