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Multifractal eigenstates of quantum chaos and the Thue-Morse sequence
Phys. Rev. E 71, 065303(R) – Published 28 June, 2005
DOI: https://doi.org/10.1103/PhysRevE.71.065303
Abstract
We analyze certain eigenstates of the quantum baker’s map and demonstrate, using the Walsh-Hadamard transform, the emergence of the ubiquitous Thue-Morse sequence, a simple sequence that is at the border between quasiperiodicity and chaos, and hence is a good paradigm for quantum chaotic states. We show a family of states that are also simply related to the Thue-Morse sequence and are strongly scarred by short periodic orbits and their homoclinic excursions. We give approximate expressions for these states and provide evidence that these and other generic states are multifractal.
Article Text
References (18)
- M. V. Berry, Proc. R. Soc. London, Ser. A 413, 183 (1987).
- E. J. Heller, Phys. Rev. Lett. 53, 1515 (1984); in Les Houches LII, Chaos and Quantum Physics, edited by M.-J. Giannoni, A. Voros, and J. Zinn-Justin (North-Holland, Amsterdam, 1991).
- F. Haake, Quantum Signatures of Chaos (Springer, Berlin, 1991).
- A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics (Springer, New York, 1992).
- J. H. Hannay and M. V. Berry, Physica D 1, 267 (1980); J. P. Keating, Nonlinearity 4, 309 (1991).
- N. L. Balazs and A. Voros, Ann. Phys. (N.Y.) 190, 1 (1989).
- M. Saraceno, Ann. Phys. (N.Y.) 199, 37 (1990).
- A. M. Ozorio de Almeida and M. Saraceno, Ann. Phys. (N.Y.) 210, 1 (1991); M. Saraceno and A. Voros, Physica D 79, 206 (1994).
- M. R. Schroeder, Number Theory in Science and Communication (Springer, Berlin, 1986).
- J.-P. Allouche and J. Shallit, in Sequences and Their Applications, Proceedings of the 1998 SETA Conference, edited by C. Ding, T. Helleseth, and H. Niederreiter (Springer, Berlin, 1999); Automatic Sequences: Theory, Applications and Generalizations (Cambridge University Press, Cambridge, 2003).
- A. Bovier and J.-M. Ghez, J. Phys. A 28, 2313 (1995).
- M. Janssen, Phys. Rep. 295, 1 (1998); A. D. Mirlin, ibid. 326, 259 (2000).
- C. Godreche and J. M. Luck, J. Phys. A 23, 3769 (1990); M. A. Zaks, A. S. Pikovsky, and J. Kurths, ibid. 32, 1523 (1999).
- T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, Phys. Rev. A 33, 1141 (1986).
- O. Penrose, Foundations of Statistical Mechanics (Pergamon Press, New York, 1970).
- R. Schack and C. M. Caves, Appl. Algebra Eng. Commun. Comput. 10, 305 (2000).
- Y. S. Weinstein, S. Lloyd, J. V. Emerson, and D. G. Cory, Phys. Rev. Lett. 89, 157902 (2002).
- A. Lakshminarayan, N. R. Cerruti, and S. Tomsovic, Phys. Rev. E 63, 016209 (2000).