Quantum resonant effects in the delta-kicked rotor revisited


We review the theoretical model and experimental realization of the atom optics δ-kicked rotor (AOKR), a paradigm of classical and quantum chaos. We have performed a number of experiments with an all-optical Bose-Einstein condensate (BEC) in a periodic standing wave potential in an AOKR system. We discuss results of the investigation of the phenomena of quantum resonances in the AOKR. An interesting feature of the momentum distribution of the atoms obtained as a result of short pulses of light, is the variance of the momentum distribution or the kinetic energy ⟨p 2⟩/2m in units of the recoil energy E rec  = ħω rec . The energy of the system is examined as a function of pulse period for a range of kicks that allow the observation of quantum resonances. In particular we study the behavior of these resonances for a large number of kicks. Higher order quantum resonant effects corresponding to the fractional Talbot time of (1/4)T T and (1/5)T T for five and ten kicks have been observed. Moreover, we describe the effect of the initial momentum of the atoms on quantum resonances in the AOKR.

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  1. 1.

    A. Lichetenberg, M. Lieberman, Regular and Chaotic Dynamics (Springer, New York, 1992)

  2. 2.

    M.A. Lieberman, A.J. Lichtenberg, Phys. Rev. A 5, 1852 (1972)

    ADS  Article  Google Scholar 

  3. 3.

    I.P. Kornfeld, S.V. Fomin, Y.G. Sinai, Ergodic Theory (Springer, New York, 1982)

  4. 4.

    B.V. Chirikov, Phys. Rep. 52, 263 (1979)

    MathSciNet  ADS  Article  Google Scholar 

  5. 5.

    G. Casati, B.V. Chirikov, J. Ford, F.M. Izraeliv, Lecture Notes in Physics (Springer, Berlin, Heidelberg, 1979)

  6. 6.

    B.V. Chirikov, F.M. Izrailev, D.L. Shepelyansky, Sov. Sci. Rev. C 2, 209 (1981)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    D.L. Shepelyanski, Physica D 8, 208 (1983)

    ADS  Article  Google Scholar 

  8. 8.

    F.L. Moore, J.C. Robinson, C.F. Bharucha, B. Sundaram, M.G. Raizen, Phys. Rev. Lett. 75, 4598 (1995)

    ADS  Article  Google Scholar 

  9. 9.

    H. Ammann, R. Gray, I. Shvarchuck, N. Christensen, Phys. Rev. Lett. 80, 4111 (1998)

    ADS  Article  Google Scholar 

  10. 10.

    H.J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, 1999)

  11. 11.

    S. Fishman, Quantum Localization, in Quantum Chaos, edited by G. Casati, I. Guarneri, U. Smilansky (North-Holland, New York, 1993), p. 187

  12. 12.

    F.M. Izrailev, D.L. Shepelyansky, Theor. Math. Phys. 43, 553 (1980)

    Article  Google Scholar 

  13. 13.

    F.M. Izrailev, Physica D 1, 243 (1980)

    MathSciNet  ADS  Article  Google Scholar 

  14. 14.

    M. Saunders, P.L. Halkyard, K.J. Challis, S.A. Gardiner, Phys. Rev. A 76, 043415 (2007)

    ADS  Article  Google Scholar 

  15. 15.

    P.L. Halkyard, M. Saunders, S.A. Gardiner, K.J. Challis, Phys. Rev. A 78, 063401 (2008)

    ADS  Article  Google Scholar 

  16. 16.

    M. Saunders, P.L. Halkyard, S.A. Gardiner, K.J. Challis, Phys. Rev. A 79, 023423 (2009)

    ADS  Article  Google Scholar 

  17. 17.

    P. McDowall, A. Hilliard, M. McGovern, T. Grünzweig, M.F. Andersen, New J. Phys. 11, 123021 (2009)

    ADS  Article  Google Scholar 

  18. 18.

    F. Saif, Phys. Rep. 419, 207 (2005)

    ADS  Article  Google Scholar 

  19. 19.

    S. Wimberger, I. Guarneri, S. Fishman, Nonlinearity 16, 1381 (2003)

    MathSciNet  ADS  Article  MATH  Google Scholar 

  20. 20.

    S. Wimberger, M. Sadgrove, J. Phys. A 38, 10549 (2005)

    MathSciNet  ADS  Article  MATH  Google Scholar 

  21. 21.

    L. Deng, E.W. Hagley, J. Denschlag, J.E. Simsarian, M. Edwards, C.W. Clark, K. Helmerson, S.L. Rolston, W.D. Phillips, Phys. Rev. Lett. 83, 5407 (1999)

    ADS  Article  Google Scholar 

  22. 22.

    M.K. Oberthaler, R.M. Godun, M.B. d’Arcy, G.S. Summy, K. Burnett, Phys. Rev. Lett. 83, 4447 (1999)

    ADS  Article  Google Scholar 

  23. 23.

    P. Szriftgiser, J. Ringot, D. Delande, J.C. Garreau, Phys. Rev. Lett. 89, 224101 (2002)

    ADS  Article  Google Scholar 

  24. 24.

    G.J. Duffy, S. Parkins, T. Müller, M. Sadgrove, R. Leonhardt, A.C. Wilson, Phys. Rev. E 70, 056206 (2004)

    ADS  Article  Google Scholar 

  25. 25.

    P.H. Jones, M. Goonasekera, D.R. Meacher, T. Jonckheere, T.S. Monteiro, Phys. Rev. Lett. 98, 073002 (2007)

    ADS  Article  Google Scholar 

  26. 26.

    S.A. Wayper, W. Simpson, M.D. Hoogerland, Eur. Phys. Lett. 79, 60006 (2007)

    ADS  Article  Google Scholar 

  27. 27.

    J.A. Currivan, A. Ullah, M.D. Hoogerland, Eur. Phys. Lett. 85, 30005 (2009)

    ADS  Article  Google Scholar 

  28. 28.

    A. Ullah, M.D. Hoogerland, Phys. Rev. E 83, 046218 (2011)

    ADS  Article  Google Scholar 

  29. 29.

    S. Fishman, D.R. Grempel, R.E. Prange, Phys. Rev. Lett. 49, 509 (1982)

    MathSciNet  ADS  Article  Google Scholar 

  30. 30.

    C. Ryu, M.F. Andersen, A. Vaziri, M.B. d’Arcy, J.M. Grossman, K. Helmerson, W.D. Phillips, Phys. Rev. Lett. 96, 160403 (2006)

    ADS  Article  Google Scholar 

  31. 31.

    Y.B. Ovchinnikov, J.H. Müller, M.R. Doery, E.J.D. Vredenbregt, K. Helmerson, S.L. Rolston, W.D. Phillips, Phys. Rev. Lett. 83, 284 (1999)

    ADS  Article  Google Scholar 

  32. 32.

    M. Lepers, V. Zehnlé, J.C. Garreau, Phys. Rev. A 77, 043628 (2008)

    ADS  Article  Google Scholar 

  33. 33.

    H.F. Talbot, Philos. Mag. 9, 401 (1836)

    Google Scholar 

  34. 34.

    M. Saunders, Ph.D. thesis, Durham University, 2009

  35. 35.

    M.P. Sadgrove, Ph.D. thesis, The University of Auckland, 2005

  36. 36.

    W.H. Oskay, D.A. Steck, V. Milner, B.G. Klappauf, M.G. Raizen, Opt. Commun. 179, 137 (2000)

    ADS  Article  Google Scholar 

  37. 37.

    M. Sadgrove, T. Mullins, S. Parkins, R. Leonhardt, Physica E 29, 369 (2005)

    ADS  Article  Google Scholar 

  38. 38.

    Y.C. Wenas, M.D. Hoogerland, Rev. Sci. Instrum. 79, 053101 (2008)

    ADS  Article  Google Scholar 

  39. 39.

    L.E. Reichl, The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations (Springer-Verlag, Berlin, 1992)

  40. 40.

    M. Berry, I. Marzoli, W. Schleich, Phys. World 14, 39 (2001)

    Google Scholar 

  41. 41.

    M. Sadgrove, M. Horikoshi, T. Sekimura, K. Nakagawa, Phys. Rev. Lett. 99, 043002 (2007)

    ADS  Article  Google Scholar 

  42. 42.

    I. Dana, V. Ramareddy, I. Talukdar, G.S. Summy, Phys. Rev. Lett. 100, 024103 (2008)

    ADS  Article  Google Scholar 

  43. 43.

    M. Sadgrove, S. Kumar, K. Nakagawa, Phys. Rev. Lett. 101, 180502 (2008)

    ADS  Article  Google Scholar 

  44. 44.

    R.A. Horne, R.H. Leonard, C.A. Sackett, Phys. Rev. A 83, 063613 (2011)

    ADS  Article  Google Scholar 

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Ullah, A., Ruddell, S., Currivan, J. et al. Quantum resonant effects in the delta-kicked rotor revisited. Eur. Phys. J. D 66, 315 (2012). https://doi.org/10.1140/epjd/e2012-30171-8

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