Sarben Sarkar
580 California St., Suite 400
San Francisco, CA, 94104
Fingerprints of classical instability in open quantum dynamics
1998, Physical Review E
https://doi.org/10.1103/PHYSREVE.58.4217
Sign up for access to the world's latest research
Abstract
The dynamics near a hyperbolic fixed point in phase space is modelled by an inverted harmonic oscillator. We investigate the effect of the classical instability on the open quantum dynamics of the oscillator, introduced through the interaction with a thermal bath, using both the survival probability function and the rate of von Neumann entropy increase, for large times. In this parameter range we prove, using influence functional techniques, that the survival probability function decreases exponentially at a rate, κ ′ , depending not only on the measure of instability in the model but also on the strength of interaction with the environment. We also show that κ ′ determines the rate of the von Neumann entropy increase and that this result is independent of the temperature of the environment. This generalises earlier results which are valid in the limit of vanishing dissipation. The validity of inferring similar rates of survival probability decrease and entropy increase for quantum chaotic systems is also discussed.
References (38)
- G. Casati and B. V. Chirikov, Quantum Chaos (Cambridge University Press, 1995).
- L. E. Reichl, The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations (Springer-Verlag, Berlin, 1992).
- A. M. Ozorio de Almeida, Hamiltonian Systems: Chaos and Quantization (Cambridge University Press, New York, 1988).
- M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, Berlin, 1990).
- F. Haake, Quantum Signatures of Chaos (Springer-Verlag, New York, 1990).
- Chaos and Quantum Physics, Les Houches Lectures Session LII, edited by M. J. Giannoni, A. Voros and J. Zinn-Justin (North Holland, Amsterdam, 1991).
- G. Casati, B. V. Chirikov, F. M. Izrailev and J. Ford, in Lecture Notes in Physics 93, edited by G. Casati and J. Ford (Springer-Verlag, Berlin, 1979).
- F. Haake, M. Kus and R. Scharf, Z. Phys. B 65, 381 (1987).
- H. J. Korsch and M. V. Berry, Physica D 3, 627 (1981).
- S. Habib, K. Shizume and W. H. Zurek, LANL preprint, quant-ph/9803042 (1998).
- S. Sarkar and J. S. Satchell, Physica D 29, 343 (1988).
- W. H. Zurek and J. P. Paz, Phys. Rev. Lett. 72, 2508 (1994);
- G. Casati and B. V. Chirikov, Phys. Rev. Lett. 75, 350 (1995);
- W. H. Zurek and J. P. Paz, Phys. Rev. Lett. 75, 351 (1995);
- W. H. Zurek and J. P. Paz, Physica D 83, 300 (1995).
- A. O. Caldeira and A. J. Leggett, Physica A 121, 587 (1983).
- D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I. O. Stamatescu and H. D. Zeh, Decoherence and the Appearance of a Classical World in Quantum Theory (Springer-Verlag, Berlin Heidelberg, 1996), and references therein; W. H. Zurek, Prog. Theor. Phys. 89, 281 (1993).
- R. Schack and C. M. Caves, Phys. Rev. E 53, 3387 (1996).
- R. Schack and C. M. Caves, Phys. Rev. E 53, 3257 (1996).
- G. Barton, Ann. Phys. (N.Y.) 166, 322 (1986).
- A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Motion (Springer-Verlag, Berlin, 1992).
- Y. B. Pesin, Russ. Math. Surveys 32 No.4, 55 (1977).
- P. Miller and S. Sarkar, preprint, submitted to Nonlinearity (1998).
- C. Beck and F. Schlögl, Thermodynamics of Chaotic Systems (Cambridge University Press, 1993).
- R. Feynman and F. Vernon, Ann. Phys. (N.Y.) 24, 118 (1963).
- B. L. Hu and A. Matacz, Phys. Rev. D 49, 6612 (1994).
- D. Koks, A. Matacz and B. L. Hu, Phys. Rev. D. 55, 5917 (1997);
- Phys. Rev. D. 56, 5281 (1997).
- E. Joos and H. D. Zeh, Z. Phys. B 59, 223 (1985).
- W. H. Zurek, LANL preprint, quant-ph/9802054 (1998).
- A. Tameshtit and J. E. Sipe, Phys. Rev. A 45, 8280 (1992);
- Phys. Rev. A 47, 1697 (1993).
- P. Pechukas, Chem. Phys. Lett. 86, 553 (1982).
- J. Wilkie and P. Brumer, Phys. Rev. Lett. 67, 1185 (1991).
- J. Pique, Y. Chen, R. Field and J. Kinsey, Phys. Rev. Lett 58, 475 (1987).
- G. M. D'Ariano, L. R. Evangelista and M. Saraceno, Phys. Rev. A 45, 3646 (1992).
- R. Zarum and S. Sarkar, Phys. Rev. E. 57, 5467 (1998).
- Handbook of Mathematical Functions, Natl. Bur. Stand. Appl. Math. Ser. No. 55, edited by M. Abramowitz and I. Stegun (U.S. GPO, Washington, DC, 1965).