Abstract
We study the time evolution ofN-level quantum systems under quasiperiodic time-dependent perturbations. The problem is formulated in terms of the spectral properties of a quasienergy operator defined in an enlarged Hilbert space, or equivalently of a generalized Floquet operator. We discuss criteria for the appearance of pure point as well as continuous spectrum, corresponding respectively to stable quasiperiodic dynamics and to unstable chaotic behavior. We discuss two types of mechanisms that lead to instability. The first one is due to near resonances, while the second one is of topological nature and can be present for arbitrary ratios between the frequencies of the perturbation. We treat explicitly an example of this type. The stability of the pure point spectrum under small perturbations is proven using KAM techniques.
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References
G. Casati, B. V. Chirikov, F. M. Izrailev, and J. Ford,Lecture Notes in Physics, Vol. 93 (Springer-Verlag, Berlin, 1979).
G. Casati and L. Molinari,Prog. Theor. Phys. Suppl. 98:287 (1989).
G. Casati and I. Guarneri,Commun. Math. Phys. 95:121 (1984).
G. A. Hagedorn, M. Loss, and J. Slawny,J. Phys. A 19:521 (1986).
M. Combescure,Ann. Inst. H. Poincaré 44:293 (1986);Ann. Phys. 173:210 (1987).
C.-A. Pillet,Commun. Math. Phys. 102:237 (1985);105:259 (1986).
M. Samuelides, R. Fleckinger, L. Touzillier, and J. Bellissard,Europhys. Lett. 1:203 (1986).
Y. Pomeau, B. Dorizzi, and B. Grammaticos,Phys. Rev. Lett. 56:681 (1986).
R. Badii and P. F. Meier,Phys. Rev. Lett. 58:1045 (1987).
P. W. Miloni, J. R. Ackerhalt, and M. E. Goggin,Phys. Rev. A 35:1714 (1987).
J. M. Luck, H. Orland, and Smilansky,J. Stat. Phys. 53:551 (1988).
D. L. Shepelyansky,Physica 8D:208 (1983).
M. Combescure,J. Stat. Phys. 62:779 (1991); and preprint, Orsay (1991).
L. Bunimovich, H. R. Jauslin, J. L. Lebowitz, A. Pellegrinotti, and P. Nielaba,J. Stat. Phys. 62:793 (1991).
J. S. Howland,Math. Ann. 207:315 (1974);Indiana Univ. Math. J. 28:471 (1979).
J. S. Howland,Ann. Inst. H. Poincaré 49:309, 325 (1989).
C. Piron,Foundations of Quantum Physics (Benjamin, Reading, Massachusetts, 1976); C. Flesia and C. Piron,Helv. Phys. Acta 57:697 (1984).
K. Yajima,Commun. Math. Phys. 87:331 (1982).
V. Enss and K. Veselic,Ann. Inst. H. Poincaré A 39:159 (1983).
J. Bellissard, inStochastic Processes in Classical and Quantum Systems, S. Albeverio, G. Casati, and D. Merlini, eds. (Springer, 1986).
H. R. Jauslin and J. L. Lebowitz,Chaos 1:114 (1991).
J. E. Bayfield and P. M. Koch,Phys. Rev. Lett. 33:258 (1974).
S. Yücel and E. Andrei,Phys. Rev. B 42:2088 (1990);43:12029 (1991).
J. Bellissard, inTrends in the Eighties, S. Albeverio and Ph. Blanchard, eds. (World Scientific, Singapore, 1985).
V. I. Arnold and A. Avez,Problèmes ergodiques de la mécanique classique (Gauthier-Villars, Paris, 1967).
R. J. Zimmer,Ergodic Theory and Semisimple Groups (Birkhäuser, Basel, 1984).
M. Rychlik, Renormalization of cocycles and linear ODE with almost-periodic coefficients,Inv. Math. (1992), to appear.
L. Amerio and G. Prouse,Almost-Periodic Functions and Functional Equations (van Nostrand Reinhold, New York, 1971).
B. M. Levitan and V. V. Zhikov,Almost-Periodic Functions and Differential Equations (Cambridge University Press, Cambridge, 1982).
H. Kesten,Acta Arithmet. 12:193 (1966).
T. Kato,Perturbation Theory for Linear Operators (Springer, Berlin, 1980).
J. Bellissard, inChaotic Behavior in Quantum Systems, G. Casati, ed. (Plenum Press, 1985).
M. Combescure,Ann. Inst. H. Poincaré 47:63 (1987);Ann. Phys. 185:86 (1988).
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Blekher, P.M., Jauslin, H.R. & Lebowitz, J.L. Floquet spectrum for two-level systems in quasiperiodic time-dependent fields. J Stat Phys 68, 271–310 (1992). https://doi.org/10.1007/BF01048846
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DOI: https://doi.org/10.1007/BF01048846