Abstract
Regular classical trajectories of dissipative systems eventually end up on limit cycles or settle on fixed points. Chaotic trajectories, on the other hand, approach so-called strange attractors whose geometry is determined by Cantor sets and their fractal dimension. In analogy with the Hamiltonian case, the two classical possibilities of simple and strange attractors are washed out by quantum fluctuations.
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Notes
- 1.
For the sake of clarity, the unit matrix is denoted by 1 here.
- 2.
For the purpose of this section, the tetradic nature of D is irrelevant; to simplify words and formulas, we shall speak of eigenvectors rather than eigen-matrices and write | 〉 rather than | 〉〉 for the basis kets.
- 3.
The subsequent discussion refers to discrete-time maps but would proceed in complete analogy for generators of infinitesimal time translations of autonomous systems.
- 4.
Throughout Sect. 12.11 the classical unit of action I∕τ will be set equal to unity. Consequently, the symbol ħ will denote Planck’s constant in units I∕τ.
- 5.
The reader should not be confused by the double meaning of Θ: While denoting the independent variable of wave-functions in (12.11.19), the symbol Θ otherwise refers to the angle coordinate of the rotator as a quantum operator.
References
U. Weiss, Quantum Dissipative Systems, 3rd edn. (World Scientific, Singapore, 2008)
D. Braun, Dissipative Quantum Chaos and Decoherence. Springer Tracts in Modern Physics, vol. 172 (Springer, Berlin, 2001)
R. Graham, F. Haake, Quantum Statistics in Optics and Solid State Physics. Springer Tracts in Modern Physics, vol. 66 (Springer, Berlin/Heidelberg, 1973)
H. Spohn, Rev. Mod. Phys. 53, 569 (1980)
F. Haake, Z. Phys. B48, 31 (1982)
F. Haake, R. Reibold, Phys. Rev. A32, 2462 (1985)
L.P. Kadanoff, G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962)
R. Bonifacio, P. Schwendimann, F. Haake, Phys. Rev. A4, 302, 854 (1971)
M. Gross, S. Haroche, Phys. Rep. 93, 301 (1982)
L.D. Landau, E.M. Lifschitz, Course of Theoretical Physics. Statistical Physics, vol. 5 (Pergamon, London, 1952)
P.A. Braun, D. Braun, F. Haake, J. Weber, Euro. Phys. J. D2, 165 (1998)
P.A. Braun, D. Braun, F. Haake, Eur. Phys. J. D3, 1 (1998)
D. Braun, P.A. Braun, F. Haake, Physica D131, 265 (1999)
D. Braun, P.A. Braun, F. Haake, Opt. Comm.179, 195 (2000); in P. Blanchard, D. Giulini, E. Joos, C. Kiefer, I.O. Stamatescu (eds.), Decoherence: Theoretical, Experimental, and Conceptual Problems (Springer, Berlin, 2000)
A.J. Leggett, Prog. Th. Phys. Suppl. 69, 80 (1980)
A.O. Caldeira, A.J. Leggett, Ann. Phys. (NY) 149, 374 (1983)
F. Haake, D.F. Walls, Phys. Rev. A36, 730 (1987)
W.H. Zurek, Phys. Rev. D24, 1516 (1981); D26, 1862 (1982); Physics Today 44, 36 (1991); Prog. Th. Phys. 89, 281 (1993)
M. Brune, E. Hagley, J. Dreyer, X. Maître, A. Maali, C. Wunderlin, J.M. Raimond, S. Haroche, Phys. Rev. Lett. 77, 4887 (1996); S. Haroche: Physics Today 51, 36 (1998)
R. Grobe, F. Haake, Z. Phys. B68, 503 (1987); Lect. Notes Phys. 282, 267 (1987)
R.R. Puri, G.S. Agarwal, Phys. Rev. A33, 3610 (1986)
S.M. Barnett, P.L. Knight, Phys. Rev. A33, 2444 (1986)
R. Grobe, F. Haake, H.-J. Sommers, Phys. Rev. Lett. 61, 1899 (1988)
L.E. Reichl, Z.-Y. Chen, M. Millonas, Phys. Rev. Lett. 63, 2013 (1989)
J. Ginibre, J. Math. Phys. 6, 440 (1965)
M.L. Mehta, Random Matrices (Academic, New York, 1967; 2nd edition 1991; 3rd edn. Elsevier, 2004)
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, New York 1970)
N. Lehmann, H.-J. Sommers, Phys. Rev. Lett. 67, 941 (1991)
R. Grobe, F. Haake, Phys. Rev. Lett. 62, 2889 (1989)
R. Grobe, Ph.D. Thesis, Essen (1989)
L. Onsager, Phys. Rev. 37, 405 (1931) and Phys. Rev. 38, 2265 (1931)
G.S. Agarwal, Z. Phys. 258, 409 (1973)
H.J. Carmichael, D.F. Walls, Z. Phys. B23, 299 (1976)
G.M. Zaslavski, Phys. Lett. 69A, 145 (1978)
G.M. Zaslavski, Kh.-R.Ya. Rachko, Sov. Phys. JETP 49, 1039 (1979)
T. Dittrich, R. Graham, Ann. Phys. 200, 363 (1990)
F.T. Arecchi, E. Courtens, G. Gilmore, H. Thomas, Phys. Rev. A6, 2211 (1972)
R. Glauber, F. Haake, Phys. Rev. A13, 357 (1976)
R.J. Glauber, Phys. Rev. 130, 2529 (1963); 131, 2766 (1963)
D.F. Walls, G. Milburn, Phys. Rev. A31, 2403 (1985)
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Haake, F., Gnutzmann, S., Kuś, M. (2018). Dissipative Systems. In: Quantum Signatures of Chaos. Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-319-97580-1_12
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