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Level Dynamics

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Quantum Signatures of Chaos

Part of the book series: Springer Series in Synergetics ((SSSYN))

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Abstract

The first attempt to understand why spectral fluctuations of quantum chaotic systems are faithfully mirrored by random matrices was proposed by Pechukas.

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Notes

  1. 1.

    It seems that in order to exhibit the Hamiltonian character of the dynamics of eigenphases and eigenvectors of the Floquet operator (11.2.2) we had to made two “educated guesses”, one concerning the choice of l mn instead of v mn, m≠n as dynamical variables, cf. (11.2.7), and the second the proper form of the Poisson brackets. In fact, both are easily derived by observing that \(F(\lambda )=\exp (-i\lambda V)F_0\) can be treated as a simple Hamiltonian flow in the space of unitary matrices. The above dynamical equations for ϕ, p, and l are then obtained by changing variables and symmetry reduction; the procedure automatically gives the l as correct variables and determines uniquely all Poisson brackets for ϕ, p, and l as well as the Hamiltonian function itself, for all symmetry classes [12, 13].

  2. 2.

    Ref. [14] relates our issue to Dirac’s theory of constrained Hamiltonian systems.

  3. 3.

    One should keep in mind that the constraint l mm = 0 can be consistently imposed only after all Poisson brackets are evaluated.

  4. 4.

    Note that we have set Planck’s constant equal to unity here such that its role is taken over by 1∕j; in more conventional notation the transition from commutators of quantum observables \(\hat {A},\hat {B},\ldots \) to Poisson brackets of the associated classical observables A, B, … reads \(\frac {\mathrm {i}}{\hbar }[\hat {A},\hat {B}]\to \{A,B\}=\frac {\partial A}{\partial p} \frac {\partial B}{\partial q}-\frac {\partial B}{\partial p}\frac {\partial A}{\partial q}\).

  5. 5.

    More precisely, over all independent real parameters in the Hermitian matrix v.

  6. 6.

    Terms with m − ν pairs plus an unmatched \(\delta (i_{k} \pm 1, i_{{\mathcal {P}} k} )\) are smaller by at least a factor 1∕N; for the sake of brevity, we forego the treatment of that case.

  7. 7.

    To simplify the appearance of T m(S, n, {l i}) we have assumed l 1≠l 2≠ … ≠l n. See the footnote to (5.12.13).

  8. 8.

    Note that the “time” τ of Sect. 11.10 is renamed as λ such that \(g=\cos \lambda \), and now the previous parameter \(\lambda =\tan \tau \) appears as \(\tan \lambda \); moreover, for the reader’s convenience, we rename the perturbation V as H 1.

References

  1. P. Pechukas, Phys. Rev. Lett. 51, 943 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  2. T. Yukawa, Phys. Rev. Lett. 54, 1883 (1985)

    Article  ADS  Google Scholar 

  3. T. Yukawa, Phys. Lett. 116A, 227 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  4. F. Haake, M. Kuś, R. Scharf, Z. Phys. B65, 381 (1987)

    Article  ADS  Google Scholar 

  5. K. Nakamura, H.J. Mikeska, Phys. Rev. A35, 5294 (1987)

    Google Scholar 

  6. H. Frahm, H.J. Mikeska, Z. Phys. B65, 249 (1986)

    Article  ADS  Google Scholar 

  7. M. Kuś, Europhys. Lett. 5, 1 (1988)

    Article  ADS  Google Scholar 

  8. O. Bohigas, M.J. Giannoni, C. Schmit, Phys. Rev. Lett. 52, 1 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  9. F. Haake, M. Kuś, Europhys. Lett. 6, 579 (1988)

    Article  ADS  Google Scholar 

  10. F. Dyson, J. Math. Phys. 3, 140 (1962)

    Article  ADS  Google Scholar 

  11. M.L. Mehta, Random Matrices (Academic, New York, 1967; 2nd edition 1991; 3rd edition Elsevier 2004)

    Google Scholar 

  12. M. Kuś, F. Haake, D. Zaitsev, A. Huckleberry, J. Phys. A 30, 8635 (1997)

    Google Scholar 

  13. A. Huckleberry, D. Zaitsev, M. Kuś, F. Haake, J. Geom. Phys. 37, 156 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  14. M. Hardej, M. Kuś, C. Gonera, P. Kosiński, J. Phys. A40, 423 (2007); for a more extended version see arXiv:nlin.cd/0608026 v1 11 Aug 2006

    ADS  Google Scholar 

  15. K. Mnich, Phys. Lett. A176, 189 (193)

    Google Scholar 

  16. P. Gaspard, S.A. Rice, H.J. Mikeska, K. Nakamura, Phys. Rev. A42, 4015 (1990)

    Article  ADS  Google Scholar 

  17. F. Calogero, C. Marchioro, J. Math. Phys. 15, 1425 (1974)

    Article  ADS  Google Scholar 

  18. B. Sutherland, Phys. Rev. A5, 1372 (1972)

    Article  ADS  Google Scholar 

  19. J. Moser, Adv. Math. 16, 1 (1975)

    Article  ADS  Google Scholar 

  20. S. Wojciechowski, Phys. Lett. 111A, 101 (1985)

    Article  ADS  Google Scholar 

  21. P.D. Lax, Comm. Pure Appl. Math. 21, 467 (1968)

    Article  MathSciNet  Google Scholar 

  22. P.A. Braun, Rev. Mod. Phys. 65, 115 (1993)

    Article  ADS  Google Scholar 

  23. P.A. Braun, S. Gnutzmann, F. Haake, M. Kuś, K. Życzkowski, Found. Phys. 31, 614 (2001)

    Article  Google Scholar 

  24. M.R. Zirnbauer, in Supersymmetry and Trace Formulae; Chaos and Disorder, ed. by I.V. Lerner, J.P. Keating, D.E. Khmelnitskii (Kluwer Academic/Plenum, New York, 1999)

    Google Scholar 

  25. E.B. Bogomolny, B. Georgeot, M.-J. Giannoni, C. Schmit, Phys. Rep. 291, 219 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  26. P. Braun, F. Haake, J. Phys. A43 262001 (2010); arXiv:1001.3339v2 [nlinCD] (2010)

    ADS  Google Scholar 

  27. P. Braun, arXiv:1508.02075v1 [nlin.CD] (2015)

    Google Scholar 

  28. J.P. Keating, Nonlinearity 4, 309 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  29. J.P. Keating, F. Mezzadri, Nonlinearity 13, 747 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  30. B. Dietz, Dissertation, Essen (1991)

    Google Scholar 

  31. B. Dietz, F. Haake, Europhys. Lett. 9, 1 (1989)

    Article  ADS  Google Scholar 

  32. F. Haake, G. Lenz, Europhys. Lett. 13, 577 (1990)

    Article  ADS  Google Scholar 

  33. D.J. Thouless, Phys. Rep. 13C, 93 (1974)

    Article  ADS  Google Scholar 

  34. P. Gaspard, S.A. Rice, H.J. Mikeska, K. Nakamura, Phys. Rev. A42, 4015 (1990)

    Article  ADS  Google Scholar 

  35. D. Saher, F. Haake, P. Gaspard, Phys. Rev. A44, 7841 (1991)

    Article  ADS  Google Scholar 

  36. B.D. Simons, A. Hashimoto, M. Courtney, D. Kleppner, B.L. Altshuler, Phys. Rev. Lett. 71, 2899 (1993)

    Article  ADS  Google Scholar 

  37. M. Kollmann, J. Stein, U. Stoffregen, H.-J. Stöckmann, B. Eckhardt, Phys. Rev. E49, R1 (1994)

    ADS  Google Scholar 

  38. D. Braun, E. Hofstetter, A. MacKinnon, G. Montambaux, Phys. Rev. B55, 7557 (1997)

    Article  ADS  Google Scholar 

  39. D. Braun, G. Montambaux, Phys. Rev. B50, 7776 (1994)

    Article  ADS  Google Scholar 

  40. J. Zakrzewski, D. Delande, Phys. Rev. E47, 1650 (1993); ibid. 1665 (1993); D. Delande, J. Zakrzewski, J. Phys. Soc. Jpn. 63, Suppl. A, 101 (1994)

    Google Scholar 

  41. F. von Oppen, Phys. Rev. Lett. 73,798 (1994); Phys. Rev. E51, 2647 (1995)

    Google Scholar 

  42. Y.V. Fyodorov, H.-J. Sommers, Phys. Rev. E51, R2719 (1995); Z. Phys. B 99, 123 (1995)

    ADS  Google Scholar 

  43. S. Iida, H.-J. Sommers, Phys. Rev. E49, 2513 (1994)

    ADS  Google Scholar 

  44. E. Akkermans, G. Montambaux, Phys. Rev. Lett. 68, 642 (1992)

    Article  ADS  Google Scholar 

  45. Y.V. Fyodorov, A.D. Mirlin, Phys. Rev. B51, 13403 (1995)

    Article  ADS  Google Scholar 

  46. M. Sieber, H. Primack, U. Smilanski, I. Ussishkin, H. Schanz, J. Phys. A28, 5041 (1995)

    ADS  Google Scholar 

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

    Book  Google Scholar 

  48. M. Barth, U. Kuhl, H.-J. Stöckmann, Phys. Rev. Lett. 82, 2026 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  49. M.V. Berry, J. Phys. A10, 2083 (1977)

    ADS  Google Scholar 

  50. V.N. Prigodin, Phys. Rev. Lett. 75, 2392 (1995)

    Article  ADS  Google Scholar 

  51. X. Yang, J. Burgdörfer, Phys. Rev. A46, 2295 (1992)

    Article  ADS  Google Scholar 

  52. B.D. Simons, B.L. Altshuler, Phys. Rev. Lett. 70, 4063 (1993); Phys. Rev. B48, 5422 (1993)

    Google Scholar 

  53. J. Zakrzewski, Z. Phys. B98, 273 (1995)

    Article  ADS  Google Scholar 

  54. I. Guarneri, K. Życzkowski, J. Zakrzewski, L. Molinari, G. Casati, Phys. Rev. E52, 2220 (1995)

    ADS  Google Scholar 

  55. G. Lenz, F. Haake, Phys. Rev. Lett. 65, 2325 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  56. G. Lenz, Dissertation (Essen, 1992)

    Google Scholar 

  57. H. Risken, The Fokker Planck Equation. Springer Series in Synergetics, vol. 18 (Springer, Berlin/Heidelberg, 1984)

    Google Scholar 

  58. A. Pandey, M.L. Mehta, Commun. Math. Phys. 87, 449 (1983)

    Article  ADS  Google Scholar 

  59. M.L. Mehta, A. Pandey, J. Phys. A16, 2655 (1983)

    Google Scholar 

  60. G. Lenz, F. Haake, Phys. Rev. Lett. 67, 1 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  61. E. Caurier, B. Grammaticos, A. Ramani, J. Phys. A23, 4903 (1990)

    ADS  Google Scholar 

  62. J.B. French, U.K.B. Kota, A. Pandey, S. Tomsovic, Ann. Phys. (N.Y.) 181, 198 (1988)

    Google Scholar 

  63. A. Pandey, Quantum Chaos and Statistical Nuclear Physics, ed. by T.H. Seligman, H. Nishioka (Springer, Berlin/Heidelberg, 1986)

    Google Scholar 

  64. F.J. Dyson, J. Math. Phys. 13, 90 (1972)

    Article  ADS  Google Scholar 

  65. L.A. Pastur, Theor. Math. Phys. 10, 67 (1972)

    Article  Google Scholar 

  66. L.A. Bunimovich, Funct. Anal. Appl. 8, 73 (1974)

    Article  Google Scholar 

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Authors and Affiliations

  1. Universität Duisburg-Essen, Essen, Germany

    Fritz Haake

  2. School of Mathematical Science, University of Nottingham, Nottingham, UK

    Sven Gnutzmann

  3. Center for Theoretical Physics, Polish Academy of Sciences, Warszawa, Poland

    Marek Kuś

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  1. Fritz Haake
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  2. Sven Gnutzmann
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  3. Marek Kuś
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Haake, F., Gnutzmann, S., Kuś, M. (2018). Level Dynamics. In: Quantum Signatures of Chaos. Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-319-97580-1_11

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