Abstract
The breaking time, known also as the Ehrenfest time, of the quantum—classical crossover scales logarithmically with respect to the Planck constant in chaotic systems. A typical dynamical system is not ergodic and not uniformly hyperbolic. Deviations from the logarithmic scale have been observed for systems with phase space structures, even in the case of strong chaos, when the homogeneous hyperbolicity of phase space does not hold. In this case the breaking time depends on scaling properties of phase space structures. Two examples of chaotic motion with di erent kind of phase space structures are presented here. The first example is quantum flights studied in the kicked rotor in the presence of the accelerator mode island structure. The second example is a model of periodically kicked harmonic oscillator with dissipation.
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References
G.P. Berman and G.M. Zaslavsky, Physica A 91, 450 (1978).
G.M. Zaslavsky, Phys. Rep. 80, 157 (1981).
G.P. Berman and A.I. Iomin, Phys. Lett. A 95, 79 (1983); Theor. Math. Phys. 77, 1197 (1988) [Teor. Mat. Fiz. 77, 277 (1988)].
D.L. Shepelyansky, Theor. Math. Phys. 49, 925 (1981) [Teor. Mat. Fiz. 49, 117 (1981)]; G.P. Berman and A.R. Kolovsky, Physica D 8, 117 (1983); V.V. Sokolov, Theor. Math. Phys. 61, 1041 (1984) [Teor. Mat. Fiz., 61, 128 (1984)].
B. V. Chirikov, F.M. Izrailev, and D.L. Shepelyansky, Sov. Sci. Rev. C2, 209 (1981).
G.P. Berman and G.M. Zaslavsky, in Quantum Chaos, editors G. Casati, B.V Chirikov. (Cambridge University Press, Cambridge, 1995), p.435.
D. Bambusi, S. Gra, and T. Paul, Asymptotic Anal. 21, 149 (1999).
R.M. Angelo, L. Sanz and K. Furuya, ibid 68, 016206 (2003); A.C. Oliveira, M.C. Nemes and K.M.F. Romero, ibid 68, 036214 (2003); Silvestrov, M.C. Goorden, and C.W.J. Beenakker, Phys. Rev. E 68, 241301 (2003).
P.G. Silvestrov and C.W.J. Beenakker, Phys. Rev. E 65, 035208 (2002); 68, 038202 (2003); S. Tomsovich and E.J. Heller, ibid 68, 038201 (2003).
See list of references in G.M. Zaslavsky, Phys. Rev. E 67 027203 (2003). [15]}
K.B. Efetov and V.R. Kogan, Phys. Rev. B 67, 245312 (2003).
I.L. Aleiner and A.I. Larkin, Phys. Rev. B 54, 14423 (1996).
Y.-C. Lai, E. Ott, and C. Grebogi, Phys. Lett. A 173, 148 (1993).
A. Iomin and G.M. Zaslavsky, Phys. Rev. E 63, 047203 (2001).
A. Iomin and G.M. Zaslavsky, Phys. Rev. E 67 027203 (2003).
J.D. Meiss, Phys. Rev. A 34, 2375 (1986); Rev. Mod. Phys. 64, 795 1992).
G.M. Zaslavsky, M. Edelman, and B.A. Niyazov, Chaos, 7, 159 (1997).
C.F.F. Karney, Physica D 8, 360 (1983).
J.D. Meiss, E. Ott, Phys. Rev. Lett. 55, 2741 (1985).
J.D. Hanson, J.R. Cary, and J.D. Meiss, J. Stat. Phys. 39, 327 (1985).
S. Fishman, D.R. Grempell and R.E. Prange, Phys. Rev. A 36, 289 (1987).
G. Casati, G. Maspero, and D.L. Shepelyansky, Phys. Rev. E 56, 6233 (1997).
D.V. Savin and V.V. Sokolov, Phys. Rev. E 56, 4911 (1997); K.M. Frahm Phys. Rev. E 56, 6237 (1997); R. Ketzmerick, Phys. Rev B 54, 10841 (1996); G. Casati, I. Guarneri, and G. Maspero, Phys. Rev. Let. 84, 63 (2000).
L. Hufnagel, R. Ketzmerick, M. Weiss, Europhys. Lett. 54, 703 (2001).
B. Sundaram and G.M. Zaslavsky, Phys. Rev. E 59, 7231 (1999).
A. Iomin and G.M. Zaslavsky, Chaos 10, 147 (2000).
R. Graham, Physica Scripta, 35, 111 (1987); T. Dittrich, R. Graham, Physica Scripta, 40, 409 (1989).
F. Haake, Quantum Signature of Chaos (Springere, Berlin, Heidelberg, 2000).
D. Braun, Dissipative Quantum Chaos and Decoherence, (Springer, Berlin, Heidelberg, 2001).
I. Percival, Quantum State Diffusion, (Cambridge University Press 1998).
T. Dittrich, in Quantum Transport and Dissipation, ed. by T. Dittrich, P. Hänggi, et. all (Wiley-VCH, Weinhem, 1998).
W.H. Zurek, Physics Today, 10, 36 (1991).
G.M. Zaslavsky, Phys. Lett. A 69, 145 (1978); G.M. Zaslavsky and Kh.-R. Ya. Rachko, Sov. Phys. JETP, 49, 1039 (1979).
Q. Wang and L.-S. Young, Commun. Math. Phys. 218, 1 (2001); 225, 275 (2002).
A.A. Vasil’ev, G.M. Zaslavsky, et. all, J.Exp.Teor.Fiz. 94, 170 (1988).
See for example M.O. Scully and M.S. Zubairy Quantum Optics, (Cambridge University Press 1997).
G.M. Zaslavsky, R.Z. Sagdeev, D.A. Usikov and A.A. Chernikov, Weak Chaos and Quasi-Regular Patterns, (Cambridge University Press 1991).
V.V. Afanas’ev, R.Z. Sagdeev, D.A. Usikov, and G.M. Zaslavsky, Phys. Let. A 152, 276 (1990); A. Iomin and G.M. Zaslavsky, Phys. Rev. E 60, 7580 (1999).
G.P. Berman, A.M. Iomin, and G.M. Zaslavsky, Physica D 4, 113 (1981).
W.H. Louisell, Radiation and Noise in Quantum Electronics (McGraw-Hill, New York, 1964).
V.K. Melnikov, in Transport, Chaos and Plasma Physics, II, Proceedings, Marseilles, edited by F. Doveil, S. Benkadda, and Y. Elskens (World Scientific, Singapore, 1996), p. 142.
G.M. Zaslavsky, Chaos 4, 25 (1994); Physica D 76, 110 (1994).
V. Rom-Kedar and G.M. Zaslavsky, Chaos, 9, 697 (1999).
B.V. Chirikov, Phys. Rep. 52, 263 (1979).
J.D. Hanson, E. Ott, and T.M. Antonsen, Phys. Rev. A 29, 1819 (1984).
S. Benkadda, S. Kassibrakis, R.B. White, and G.M. Zaslavsky, Phys. Rev. E 55, 4907 (1997).
A. Iomin and G.M. Zaslavsky, Chem. Phys. 284, 3 (2002).
G. Casati, G. Maspero, and D.L. Shepelyansky, Phys. Rev. Lett. 82, 524 (1999).
R. Ketzmerick, L. Hufnagel, F. Steinbah, and M. Weiss, Phys. Rev. Lett. 85, 1214 (2000)
P. Carruthers and M.M. Nieto, Rev. Mod. Phys. 40, 411 (1968).
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Iomin, A., Zaslavsky, G. (2005). Quantum Breaking Time for Chaotic Systems with Phase Space Structures. In: Collet, P., Courbage, M., Métens, S., Neishtadt, A., Zaslavsky, G. (eds) Chaotic Dynamics and Transport in Classical and Quantum Systems. NATO Science Series, vol 182. Springer, Dordrecht. https://doi.org/10.1007/1-4020-2947-0_15
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DOI: https://doi.org/10.1007/1-4020-2947-0_15
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