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Exploiting Chaos for Quantum Control

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International Conference on Theory and Application in Nonlinear Dynamics (ICAND 2012)

Part of the book series: Understanding Complex Systems ((UCS))

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Abstract

The field of Quantum Chaos is referred to as the study of quantum behaviors of systems whose corresponding classical dynamics are chaotic, or study of quantum manifestations of classical chaos. Equivalently, it means that quantum behaviors depend on the nature of the classical dynamics, implying that classical chaos can be used to control or manipulate quantum behaviors. We discuss two examples here: using transient chaos to control quantum transport in nanoscale systems and exploiting chaos to regularize relativistic quantum tunneling dynamics in Dirac fermion and graphene systems.

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References

  1. E. Ott, C. Grebogi, J.A. Yorke, Phys. Rev. Lett. 64, 1196 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Boccaletti, C. Grebogi, Y.-C. Lai, H. Mancini, D. Maza, Phys. Rep. 329, 103 (2000)

    Article  MathSciNet  Google Scholar 

  3. L.M. Pecora, H. Lee, D.-H. Wu, T. Antonsen, M.-J. Lee, E. Ott, Phys. Rev. E 83, 065201 (2011)

    Article  Google Scholar 

  4. R. Yang, L. Huang, Y.-C. Lai, L.M. Pecora, Appl. Phys. Lett. 100, 093105 (2012)

    Article  Google Scholar 

  5. X. Ni, L. Huang, Y.-C. Lai, L.M. Pecora, EPL 98, 50007 (2012)

    Article  Google Scholar 

  6. S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, Cambridge, 1995)

    Book  Google Scholar 

  7. See, for example, R. A. Jalabert, H. U. Baranger, and A. D. Stone, Phys. Rev. Lett. 65, 2442 (1990)

    Google Scholar 

  8. R. Ketzmerick, Phys. Rev. B 54, 10841 (1996)

    Article  Google Scholar 

  9. R.P. Taylor, R. Newbury, A.S. Sachrajda, Y. Feng, P.T. Coleridge, C. Dettmann, N. Zhu, H. Guo, A. Delage, P. J. Kelly, Z. Wasilewski. Phys. Rev. Lett. 78, 1952 (1997)

    Google Scholar 

  10. A.S. Sachrajda, R. Ketzmerick, C. Gould, Y. Feng, P.J. Kelly, A. Delage, Z. Wasilewski, Phys. Rev. Lett. 80, 1948 (1998)

    Article  Google Scholar 

  11. B. Huckestein, R. Ketzmerick, C.H. Lewenkopf, Phys. Rev. Lett. 84, 5504 (2000)

    Article  Google Scholar 

  12. G. Casati, I. Guarneri, G. Maspero, Phys. Rev. Lett. 84, 63 (2000)

    Article  Google Scholar 

  13. R. Crook, C.G. Smith, A.C. Graham, I. Farrer, H.E. Beere, D.A. Ritchie, Phys. Rev. Lett. 91, 246803 (2003)

    Article  Google Scholar 

  14. W.H. Zurek, Rev. Mod. Phys. 75, 715 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Akis, J.P. Bird, D.K. Ferry, Appl. Phys. Lett. 81, 129 (2002)

    Article  Google Scholar 

  16. D.K. Ferry, R. Akis, J.P. Bird, Phys. Rev. Lett. 93, 026803 (2004)

    Article  Google Scholar 

  17. U. Fano, Phys. Rev. 124, 1866 (1961)

    Article  MATH  Google Scholar 

  18. H. Ishio, Phys. Rev. E 62, R3035 (2000)

    Article  Google Scholar 

  19. Given a closed Hamiltonian system that exhibits fully developed chaos in the classical limit, one might expect the quantum wavefunctions associated with various eigenstates to be more or less uniform in the physical space. However, in the seminal work of McDonald and Kaufman [Phys. Rev. Lett. 42, 1189 (1979) and Phys. Rev. A 37, 3067 (1988)], it was observed that quantum eigen-wavefunctions can be highly non-uniform in the chaotic stadium billiard. A systematic study was subsequently carried out by Heller [Phys. Rev. Lett. 53, 1515 (1984)], who established the striking tendency for wavefunctions to concentrate about classical unstable periodic orbits, which he named quantum scars. Semiclassical theory was then developed by Bogomolny [Physica D 31, 169 (1988)] and Berry [Proc. Roy. Soc. (London) A 423, 219 (1989)], providing a general understanding of the physical mechanism of quantum scars. The phenomenon of quantum scarring was deemed counterintuitive and surprising but only for chaotic systems, as the phase space of an integrable system is not ergodic so that the quantum wavefunctions are generally not expected to be uniform. Relativistic quantum scars in chaotic graphene systems have also been reported [L. Huang, Y.-C. Lai, D. K. Ferry, S. M. Goodnick, and R. Akis, Phys. Rev. Lett. 103, 054101 (2009)].

    Google Scholar 

  20. Y.-C. Lai, T. Tél, Transient Chaos (Springer, New York, 2011)

    Book  MATH  Google Scholar 

  21. Y. Sinai, Russ. Math. Surv. 25, 137 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  22. See, for example, Chapter 18 in J. R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics (Cambridge University Press, Cambridge 1999)

    Google Scholar 

  23. K.S. Novoselov et al., Science 306, 666 (2004)

    Article  Google Scholar 

  24. C. Berger et al., J. Phys. Chem. B 108, 19912 (2004)

    Article  Google Scholar 

  25. C.W.J. Beenakker, Rev. Mod. Phys. 80, 1337 (2008)

    Article  Google Scholar 

  26. A.H. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009)

    Article  Google Scholar 

  27. R. Blümel, U. Smilansky, Phys. Rev. Lett. 60, 477 (1988)

    Article  Google Scholar 

  28. R. Blümel, U. Smilansky, Physica D 36, 111 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  29. Y.-C. Lai, R. Blümel, E. Ott, C. Grebogi, Phys. Rev. Lett. 68, 3491 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  30. R. Yang, L. Huang, Y.-C. Lai, C. Grebogi, and L. M. Pecora, Chaos 23, 013125 (2013)

    Google Scholar 

  31. X. Ni, L. Huang, Y.-C. Lai, C. Grebogi, Phys. Rev. E 86, 015702 (2012)

    Google Scholar 

  32. P. Strange, Relativistic Quantum Mechanics with Applications in Condensed Matter Physics and Atomic Physics (Cambridge University Press, Cambridge, 1998)

    Book  Google Scholar 

  33. M.I. Katsnelson, K.S. Novoselov, A.K. Geim, Nat. Phys. 2, 620 (2006)

    Article  Google Scholar 

  34. C.W.J. Beenakker, Colloquium: andreev reflection and Klein tunneling in graphene. Rev. Mod. Phys. 80, 1337 (2008)

    Article  Google Scholar 

  35. F. Haake, Quantum Signatures of Chaos, 2nd edn. (Springer, Berlin, 2001)

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  37. W. Breymann, Z. Kov’acs, T. Tél, Phys. Rev. E 50, 1994 (1994)

    Article  Google Scholar 

  38. G.-L. Wang, L. Ying, Y.-C. Lai, and C. Grebogi, Phys. Rev. E 87, 052908 (2013)

    Google Scholar 

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Acknowledgments

The main idea of using chaos to manipulate quantum behaviors was generated through extensive discussions with Dr. L. Pecora from Naval Research Laboratory in January 2011 at Dr. M. Shlesinger’s ONR Program Review Meeting at UCSD. The computations and theoretical analyses reported in the references [4, 5, 30, 31] on which this Review is based were mainly carried out by Dr. R. Yang, Dr. X. Ni, and Dr. L. Huang, all formerly affiliated with ASU.

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Authors and Affiliations

  1. School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona,  85287, USA

    Ying-Cheng Lai

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  1. Ying-Cheng Lai
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Correspondence to Ying-Cheng Lai .

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Editors and Affiliations

  1. Code 71730, Spawar Systems Center, San Diego, California, USA

    Visarath In

  2. Department of Mathematics, San Diego State University, San Diego, California, USA

    Antonio Palacios

  3. Code 71730, Spawar Systems Center, San Diego, California, USA

    Patrick Longhini

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Lai, YC. (2014). Exploiting Chaos for Quantum Control. In: In, V., Palacios, A., Longhini, P. (eds) International Conference on Theory and Application in Nonlinear Dynamics (ICAND 2012). Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-02925-2_1

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