Nonlinear quantum evolution with maximal entropy production

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 63, Issue 2, 2001, Pages 022105-022101

  Gheorghiu Svirschevski, S ^a  

^a NONE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; ENERGY CONSERVATION; ENTROPY; HAMILTONIANS; PROBABILITY DISTRIBUTIONS; RELAXATION PROCESSES; THERMODYNAMIC STABILITY;

NON-LINEAR QUANTUM EVOLUTION;

QUANTUM THEORY;

EID: 0035251021     PISSN: 10502947     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevA.63.022105     Document Type: Article

References (36)

1. J.J. Bollinger et al., Phys. Rev. Lett. 63, 1031 (1989); R.L. Walsworth, F. Silvera, E.M. Mattison, and R.F.C. Vessot, ibid. 64, 2599 (1990); T.E. Chupp and R.J. Hoare, ibid. 64, 2261 (1990); P.K. Majumder et al., ibid. 65, 2931 (1990); F. Benatti and R. Floreanini, Phys. Lett. B 389, 100 (1996); 451, 422 (1999); E. Lisi, A. Marrone, and D. Montanino, Phys. Rev. Lett. 85, 1166 (2000); e-print hep-ph/0002053; H.V. Klapdor-Kleingrothaus, H. Pas, and U. Sarkar, e-print hep-ph/0004123.
2. J.J. Bollinger et al., Phys. Rev. Lett. 63, 1031 (1989); R.L. Walsworth, F. Silvera, E.M. Mattison, and R.F.C. Vessot, ibid. 64, 2599 (1990); T.E. Chupp and R.J. Hoare, ibid. 64, 2261 (1990); P.K. Majumder et al., ibid. 65, 2931 (1990); F. Benatti and R. Floreanini, Phys. Lett. B 389, 100 (1996); 451, 422 (1999); E. Lisi, A. Marrone, and D. Montanino, Phys. Rev. Lett. 85, 1166 (2000); e-print hep-ph/0002053; H.V. Klapdor-Kleingrothaus, H. Pas, and U. Sarkar, e-print hep-ph/0004123.
3. J.J. Bollinger et al., Phys. Rev. Lett. 63, 1031 (1989); R.L. Walsworth, F. Silvera, E.M. Mattison, and R.F.C. Vessot, ibid. 64, 2599 (1990); T.E. Chupp and R.J. Hoare, ibid. 64, 2261 (1990); P.K. Majumder et al., ibid. 65, 2931 (1990); F. Benatti and R. Floreanini, Phys. Lett. B 389, 100 (1996); 451, 422 (1999); E. Lisi, A. Marrone, and D. Montanino, Phys. Rev. Lett. 85, 1166 (2000); e-print hep-ph/0002053; H.V. Klapdor-Kleingrothaus, H. Pas, and U. Sarkar, e-print hep-ph/0004123.
4. J.J. Bollinger et al., Phys. Rev. Lett. 63, 1031 (1989); R.L. Walsworth, F. Silvera, E.M. Mattison, and R.F.C. Vessot, ibid. 64, 2599 (1990); T.E. Chupp and R.J. Hoare, ibid. 64, 2261 (1990); P.K. Majumder et al., ibid. 65, 2931 (1990); F. Benatti and R. Floreanini, Phys. Lett. B 389, 100 (1996); 451, 422 (1999); E. Lisi, A. Marrone, and D. Montanino, Phys. Rev. Lett. 85, 1166 (2000); e-print hep-ph/0002053; H.V. Klapdor-Kleingrothaus, H. Pas, and U. Sarkar, e-print hep-ph/0004123.
5. J.J. Bollinger et al., Phys. Rev. Lett. 63, 1031 (1989); R.L. Walsworth, F. Silvera, E.M. Mattison, and R.F.C. Vessot, ibid. 64, 2599 (1990); T.E. Chupp and R.J. Hoare, ibid. 64, 2261 (1990); P.K. Majumder et al., ibid. 65, 2931 (1990); F. Benatti and R. Floreanini, Phys. Lett. B 389, 100 (1996); 451, 422 (1999); E. Lisi, A. Marrone, and D. Montanino, Phys. Rev. Lett. 85, 1166 (2000); e-print hep-ph/0002053; H.V. Klapdor-Kleingrothaus, H. Pas, and U. Sarkar, e-print hep-ph/0004123.
6. J.J. Bollinger et al., Phys. Rev. Lett. 63, 1031 (1989); R.L. Walsworth, F. Silvera, E.M. Mattison, and R.F.C. Vessot, ibid. 64, 2599 (1990); T.E. Chupp and R.J. Hoare, ibid. 64, 2261 (1990); P.K. Majumder et al., ibid. 65, 2931 (1990); F. Benatti and R. Floreanini, Phys. Lett. B 389, 100 (1996); 451, 422 (1999); E. Lisi, A. Marrone, and D. Montanino, Phys. Rev. Lett. 85, 1166 (2000); e-print hep-ph/0002053; H.V. Klapdor-Kleingrothaus, H. Pas, and U. Sarkar, e-print hep-ph/0004123.
7. J.J. Bollinger et al., Phys. Rev. Lett. 63, 1031 (1989); R.L. Walsworth, F. Silvera, E.M. Mattison, and R.F.C. Vessot, ibid. 64, 2599 (1990); T.E. Chupp and R.J. Hoare, ibid. 64, 2261 (1990); P.K. Majumder et al., ibid. 65, 2931 (1990); F. Benatti and R. Floreanini, Phys. Lett. B 389, 100 (1996); 451, 422 (1999); E. Lisi, A. Marrone, and D. Montanino, Phys. Rev. Lett. 85, 1166 (2000); e-print hep-ph/0002053; H.V. Klapdor-Kleingrothaus, H. Pas, and U. Sarkar, e-print hep-ph/0004123.
8. J.J. Bollinger et al., Phys. Rev. Lett. 63, 1031 (1989); R.L. Walsworth, F. Silvera, E.M. Mattison, and R.F.C. Vessot, ibid. 64, 2599 (1990); T.E. Chupp and R.J. Hoare, ibid. 64, 2261 (1990); P.K. Majumder et al., ibid. 65, 2931 (1990); F. Benatti and R. Floreanini, Phys. Lett. B 389, 100 (1996); 451, 422 (1999); E. Lisi, A. Marrone, and D. Montanino, Phys. Rev. Lett. 85, 1166 (2000); e-print hep-ph/0002053; H.V. Klapdor-Kleingrothaus, H. Pas, and U. Sarkar, e-print hep-ph/0004123.
9. S.W. Hawking, Phys. Rev. D 14, 2460 (1976); J. Ellis, J.S. Hagelin, D.V. Nanopoulos, and M. Srednicki, Nucl. Phys. B 241, 381 (1984); T. Banks, L. Suskind, and M.E. Peskin, ibid. 244, 125 (1984).
10. S.W. Hawking, Phys. Rev. D 14, 2460 (1976); J. Ellis, J.S. Hagelin, D.V. Nanopoulos, and M. Srednicki, Nucl. Phys. B 241, 381 (1984); T. Banks, L. Suskind, and M.E. Peskin, ibid. 244, 125 (1984).
11. S.W. Hawking, Phys. Rev. D 14, 2460 (1976); J. Ellis, J.S. Hagelin, D.V. Nanopoulos, and M. Srednicki, Nucl. Phys. B 241, 381 (1984); T. Banks, L. Suskind, and M.E. Peskin, ibid. 244, 125 (1984).
12. W.G. Unruh and R.M. Wald, Phys. Rev. D 52, 2176 (1995).
13. S.B. Giddings and A. Strominger, Nucl. Phys. B 307, 854 (1988); J. Ellis, N.B. Mavromatos, and D. Nanopoulos, Phys. Lett. B 293, 142 (1992); P. Huet and M.E. Peskin, Nucl. Phys. B 434, 3 (1995); J. Ellis, J.L. Lopez, N.E. Mavromatos, and D. Nanopoulos, Phys. Rev. D 53, 3846 (1996); C.-H. Chang et al., e-print hep-ph/9711310; F. Benatti and R. Floreanini, J. High Energy Phys. 2, 32 (2000).
14. S.B. Giddings and A. Strominger, Nucl. Phys. B 307, 854 (1988); J. Ellis, N.B. Mavromatos, and D. Nanopoulos, Phys. Lett. B 293, 142 (1992); P. Huet and M.E. Peskin, Nucl. Phys. B 434, 3 (1995); J. Ellis, J.L. Lopez, N.E. Mavromatos, and D. Nanopoulos, Phys. Rev. D 53, 3846 (1996); C.-H. Chang et al., e-print hep-ph/9711310; F. Benatti and R. Floreanini, J. High Energy Phys. 2, 32 (2000).
15. S.B. Giddings and A. Strominger, Nucl. Phys. B 307, 854 (1988); J. Ellis, N.B. Mavromatos, and D. Nanopoulos, Phys. Lett. B 293, 142 (1992); P. Huet and M.E. Peskin, Nucl. Phys. B 434, 3 (1995); J. Ellis, J.L. Lopez, N.E. Mavromatos, and D. Nanopoulos, Phys. Rev. D 53, 3846 (1996); C.-H. Chang et al., e-print hep-ph/9711310; F. Benatti and R. Floreanini, J. High Energy Phys. 2, 32 (2000).
16. S.B. Giddings and A. Strominger, Nucl. Phys. B 307, 854 (1988); J. Ellis, N.B. Mavromatos, and D. Nanopoulos, Phys. Lett. B 293, 142 (1992); P. Huet and M.E. Peskin, Nucl. Phys. B 434, 3 (1995); J. Ellis, J.L. Lopez, N.E. Mavromatos, and D. Nanopoulos, Phys. Rev. D 53, 3846 (1996); C.-H. Chang et al., e-print hep-ph/9711310; F. Benatti and R. Floreanini, J. High Energy Phys. 2, 32 (2000).
17. S.B. Giddings and A. Strominger, Nucl. Phys. B 307, 854 (1988); J. Ellis, N.B. Mavromatos, and D. Nanopoulos, Phys. Lett. B 293, 142 (1992); P. Huet and M.E. Peskin, Nucl. Phys. B 434, 3 (1995); J. Ellis, J.L. Lopez, N.E. Mavromatos, and D. Nanopoulos, Phys. Rev. D 53, 3846 (1996); C.-H. Chang et al., e-print hep-ph/9711310; F. Benatti and R. Floreanini, J. High Energy Phys. 2, 32 (2000).
18. S.B. Giddings and A. Strominger, Nucl. Phys. B 307, 854 (1988); J. Ellis, N.B. Mavromatos, and D. Nanopoulos, Phys. Lett. B 293, 142 (1992); P. Huet and M.E. Peskin, Nucl. Phys. B 434, 3 (1995); J. Ellis, J.L. Lopez, N.E. Mavromatos, and D. Nanopoulos, Phys. Rev. D 53, 3846 (1996); C.-H. Chang et al., e-print hep-ph/9711310; F. Benatti and R. Floreanini, J. High Energy Phys. 2, 32 (2000).
19. G. Domokos and S. Kovesi-Domokos, J. Phys. A 32, 4105 (1999).
20. T.F. Jordan, Ann. Phys. (N.Y.) 225, 83 (1993). Further details on the class of Lie-Poisson dynamics and its interesting properties can be found in M. Czachor, M. Kuna, S. B. Leble, and J. Naudts, in Trends in Quantum Mechanics, edited by H.-D. Doebner et al. (World Scientific, Singapore, 2000); M. Czachor, Phys. Rev. A 57, 4122 (1998); M. Czachor and M. Kuna, ibid. 58, 128 (1998); S.B. Leble and M. Czachor, Phys. Rev. E 58, 7091 (1998); M. Czachor and J. Naudts, ibid. 59, 2497 (1999).
21. T.F. Jordan, Ann. Phys. (N.Y.) 225, 83 (1993). Further details on the class of Lie-Poisson dynamics and its interesting properties can be found in M. Czachor, M. Kuna, S. B. Leble, and J. Naudts, in Trends in Quantum Mechanics, edited by H.-D. Doebner et al. (World Scientific, Singapore, 2000); M. Czachor, Phys. Rev. A 57, 4122 (1998); M. Czachor and M. Kuna, ibid. 58, 128 (1998); S.B. Leble and M. Czachor, Phys. Rev. E 58, 7091 (1998); M. Czachor and J. Naudts, ibid. 59, 2497 (1999).
22. T.F. Jordan, Ann. Phys. (N.Y.) 225, 83 (1993). Further details on the class of Lie-Poisson dynamics and its interesting properties can be found in M. Czachor, M. Kuna, S. B. Leble, and J. Naudts, in Trends in Quantum Mechanics, edited by H.-D. Doebner et al. (World Scientific, Singapore, 2000); M. Czachor, Phys. Rev. A 57, 4122 (1998); M. Czachor and M. Kuna, ibid. 58, 128 (1998); S.B. Leble and M. Czachor, Phys. Rev. E 58, 7091 (1998); M. Czachor and J. Naudts, ibid. 59, 2497 (1999).
23. T.F. Jordan, Ann. Phys. (N.Y.) 225, 83 (1993). Further details on the class of Lie-Poisson dynamics and its interesting properties can be found in M. Czachor, M. Kuna, S. B. Leble, and J. Naudts, in Trends in Quantum Mechanics, edited by H.-D. Doebner et al. (World Scientific, Singapore, 2000); M. Czachor, Phys. Rev. A 57, 4122 (1998); M. Czachor and M. Kuna, ibid. 58, 128 (1998); S.B. Leble and M. Czachor, Phys. Rev. E 58, 7091 (1998); M. Czachor and J. Naudts, ibid. 59, 2497 (1999).
24. T.F. Jordan, Ann. Phys. (N.Y.) 225, 83 (1993). Further details on the class of Lie-Poisson dynamics and its interesting properties can be found in M. Czachor, M. Kuna, S. B. Leble, and J. Naudts, in Trends in Quantum Mechanics, edited by H.-D. Doebner et al. (World Scientific, Singapore, 2000); M. Czachor, Phys. Rev. A 57, 4122 (1998); M. Czachor and M. Kuna, ibid. 58, 128 (1998); S.B. Leble and M. Czachor, Phys. Rev. E 58, 7091 (1998); M. Czachor and J. Naudts, ibid. 59, 2497 (1999).
25. T.F. Jordan, Ann. Phys. (N.Y.) 225, 83 (1993). Further details on the class of Lie-Poisson dynamics and its interesting properties can be found in M. Czachor, M. Kuna, S. B. Leble, and J. Naudts, in Trends in Quantum Mechanics, edited by H.-D. Doebner et al. (World Scientific, Singapore, 2000); M. Czachor, Phys. Rev. A 57, 4122 (1998); M. Czachor and M. Kuna, ibid. 58, 128 (1998); S.B. Leble and M. Czachor, Phys. Rev. E 58, 7091 (1998); M. Czachor and J. Naudts, ibid. 59, 2497 (1999).
26. S. R. De Groot and P. Mazur, Non-equilibrium Thermodynamics (Dover, New York, 1984).
27. I. Bialynicki-Birula and J. Mycielski, Ann. Phys. (N.Y.) 100, 62 (1976); J. Polchinski, Phys. Rev. Lett. 66, 397 (1991).
28. I. Bialynicki-Birula and J. Mycielski, Ann. Phys. (N.Y.) 100, 62 (1976); J. Polchinski, Phys. Rev. Lett. 66, 397 (1991).
29. C. Tsallis, J. Stat. Phys. 52, 479 (1988); C. Tsallis, R.S. Mendes, and A.R. Plastino, Physica A 261, 534 (1998), and references therein.
30. C. Tsallis, J. Stat. Phys. 52, 479 (1988); C. Tsallis, R.S. Mendes, and A.R. Plastino, Physica A 261, 534 (1998), and references therein.
31. B. Reznik, Phys. Rev. Lett. 76, 1192 (1996).
32. To our knowledge the tilde conjugate was first proposed in the formalism of thermofield dynamics; see, e.g., A. Arimitsu and H. Umezawa, Prog. Theor. Phys. 74, 429 (1985). There it was defined by (A|α〉)†=(Ã)†|α†〉. Our definition in Eq. (5) has the advantage of being applicable to nonlinear superoperators as well; it is also instrumental in giving a concise form to certain superoperator relations, e.g., the first dissipativity conv dition of Lindblad [12] becomes simply A=Ã, i.e., the condition of tilde symmetry explained in the text; also see Sec. IV.
33. G. Lindblad, Commun. Math. Phys. 48, 119 (1976).
34. Handbook of Mathematical Functions, edited by M. Abramovitz and I. A. Stegun (Dover, New York, 1970).
35. N. Gisin, J. Phys. A 14, 2259 (1981); Physica A 111, 364 (1982).
36. N. Gisin, J. Phys. A 14, 2259 (1981); Physica A 111, 364 (1982).