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Recurrent versus diffusive dynamics for a kicked quantum system. (English) Zbl 0746.11014
The author studies the dynamics of a two-level quantum system subject to a time-dependent kicking perturbation modulated along the Prouhet-Thue- Morse sequence: the Hamiltonian she studies is given by: \[ H(t)=E\sigma_ z + \lambda\sigma_ x \sum^ \infty_{- \infty}\gamma_ n\delta(t-n), \] where \(E\) and \(\lambda\) are real parameters, \(\sigma_ x\) and \(\sigma_ z\) are the Pauli matrices \(\sigma_ x = \left({0\atop 1}{1\atop 0}\right)\), \(\sigma_ z = \left({1\atop 0}{0\atop -1}\right)\), and \((\gamma_ n)\) is the two-sided Thue-Morse sequence. The author proves that, for a nontrivial set of parameters \(E\), \(\lambda\) (that she conjectures to be a dense set), the quantum dynamics is both recurrent and diffusive. The quantum autocorrelation function (for any initial state) is the Fourier transform of a measure with a pure point part and a singular continuous part (the latter being closely related to the Fourier transform of the Thue-Morse sequence).

MSC:
11B85 Automata sequences
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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[1] J. P. Allouche and J. Peyriére,C. R. Acad. Sci. Paris 302:1135 (1986).
[2] F. Axel and J. Peyriére,J. Stat. Phys. 57:1013 (1989). · Zbl 0724.11011
[3] J. Bellissard, inLecture Notes in Mathematical Physics, No. 1049 (Springer, 1985).
[4] J. Bellissard, inThéorie des Nombres et Physique (Springer Proceedings in Physics, Vol.47), J. M. Luck, P. Moussa, and M. Waldschmidt, eds. (Springer, Berlin, 1990).
[5] R. Blümel and U. Smilansky,Phys. Rev. Lett. 58:2531 (1987).
[6] G. Casati and I. Guarneri,Commun. Math. Phys. 95:12 (1984). · Zbl 0581.46062
[7] G. Casati, B. V. Chirikov, D. L. Shepelyanski, and I. Guarneri,Phys. Rep. 154:77 (1987).
[8] A. Cohen and S. Fishman,Int. J. Mod. Phys. B 2:103 (1988).
[9] M. Combescure,Ann. Inst. Henri Poincaré 47:63 (1987).
[10] M. Combescure,Ann. Phys. 185:86 (1988). · Zbl 0655.35076
[11] M. Combescure,J. Stat. Phys. 59 (1990).
[12] B. Eckardt,Phys. Rep. 163:3590 (1988).
[13] S. Fishman, D. R. Grempel, and R. E. Prange,Phys. Rev. A 36:289 (1987).
[14] T. Geisel,Phys. Rev. A 41:2989 (1990).
[15] I. Guarneri,Lett. Nuovo Cimento 40:171 (1984).
[16] F. Haake, M. Kus, and R. Scharf,Z. Phys. B 65:381 (1987).
[17] J. Howland,Indiana Math. J. 28:471 (1979). · Zbl 0444.47010
[18] J. Howland,Ann. Inst. H. Poincaré A 50:309 (1989).
[19] J. Howland,Ann. Inst. H. Poincaré A 50:325 (1989).
[20] S. Kotani,Rev. Math. Phys. 1:129 (1989). · Zbl 0713.60074
[21] J. M. Luck, H. Orland, and U. Smilansky,J. Stat. Phys. 53:551 (1988).
[22] J. M. Luck,Phys. Rev. B 39:5834 (1989).
[23] B. Milek and P. Seba, Singular continuous quasi energy spectrum in the kicked rotator with separable perturbation: Onset of quantum chaos, Preprint, Ruhr Universität Bochum (1989).
[24] J. Peyriére, private communication.
[25] J. Piepenbrink, private communication.
[26] M. Queflelec,Substitution Dynamical Systems; Spectral Analysis (Springer, Berlin, 1987).
[27] E. Shimshoni and U. Smilansky,Nonlinearly 1:435 (1988). · Zbl 0662.76158
[28] Y. Sinai, Mathematical problems in the theory of quantum chaos, Preprint, Moscow (1989). · Zbl 0748.01004
[29] B. Sutherland,Phys. Rev. Lett. 57:770 (1986).
[30] M. Kolar, M. K. Ali, and F. Nori, Generalized Thue Morse chains and their properties,Phys. Rev. B. to appear.
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