Long-time transients in open quantum systems

1. Long-time transients in open quantum systems Krzysztof Stefański CM UMK Bydgoszcz/Toruń 44th SMP, Toruń 2012

2. Transient behaviour – what does one mean? A behaviour of a dynamical system that is qualitatively different from its asymptotic (”eventual”) behaviour.Such a behaviour can be observed exclusively in dissipative systems!

3. Transient behaviours in classical open systems An example – transient chaos generated by maps with periodic attractors f : I→ I, xn+1 = f(xn), e.g. Logistic maps, where f(x) = rx(1-x)

4. Bifurcation diagram for the family of Logistic maps

5. A vicinity of a period-5 window after 100 iterations

6. The same after 1000 iterations

7. Distribution of rambling times in a sample of 106 trajectories 1000000

8. Same in the semi-log scale

9. 3-D maps generating hyperchaotic trajectories:

10. A hyperchaotic attractor generated by an F2 map

11. Disitribution of rambling times for a sample of 250 trajectories in the period-3 window (RT-axis unit: 106 iterations; the average is ca 0.7• 106(!)) [1]

12. Transient chaos in classical systems can be characterized by time evolution of the average finite-time estimates of Lyapunov exponent(s). For a typical set of trajectories one can observe a continuous decrease of the estimates and their convergence to the „true” Lyapunov exponents.

13. Time evolution of finite-timeestimates of LE in a periodic window

14. Comparison of the model and nuericaly obtained time evolutions of AFTLE for a periodic window of the family of Logistic maps. [2]

15. And the quantum systems? 1. It may be a bit difficult to find a nondisputable equivalent of transient chaos. 2. However, one can look for various examples of quantum metastability.

16. A bit of algebra Fokker-Planck equation [3]: May be derived from the Langevine equation or Liouville-von Neumann equation

17. Expansion of P leads to the Schroedinger-like equation where [3]

18. A schematic draft of the typical shape of the potential U in the case of bistability is shown below. Two wells with negative minima are separated by a barrier with an additional, positive-value minimum. Height of the barrier increases with the size of the system N(or, in general, with decrese of the amplitude of diffusion function q)

21. Such a potential implies that the two lowest eigenvalues λ0 = 0, and λ1 are close to each other, the closer, the higher the barrier (quasi- or near-degeneracy). The corresponding eigenfunctions are almost completely localized over the wells and differ mainly in that p1 has a node:

22. Schematic plot of eigenfunctions p0 for two various amplitudes of diffusion function q (or the size of the system N)

23. Same for the eigenfunctions p1

24. Long time evolution of the system can be represented with the two eigenfunctions p0 & p1 only, since the remaining components of the expansion of P decay rapidly and do not count except for the very initial time interval Δτdefined by the formula Δτ = k/λ2, where k follows from the required level of accuracy of the approximation: P(x,t) ≈ A0χ(x)p0(x) + A1χ(x)p1(x)exp(- λ1t).

25. Eigenfunction p0 for theeigenvalue λ0

26. Eigenfunction p1 for theeigenvalue λ1

27. For times fulfilling the double inequality: k/λ2 < t≪ 1/λ1, instead of p0 and p1, shown above, one can use another pair of quasi-eigenfunctions pL & pR, that are shown below:

28. Pseudo-eigenfunction pL

29. Pseudo-eigenfunction pR

30. Two comments: • In the quantum Hamiltonian eigenproblems with a double-well potential one obtains almost identical eigenfunctions but their physical meaning is essentially different leaving n room for transients. • In the case of classical transient chaos trajectories while approaching the attractor have to ”tunnel” through a labyrinth-like ”pesudobarrier”.

31. Conclusions (?) • Laser with saturable absorber (or dye laser) provides a simple model for quantum tunneling and, consequently, transient behavior in essentially quantum systems. • Large molecules (e.g. proteins) may be even more interesting and potentially important (though difficult) subjects for similar treatment and analysis.

32. Th.Y.f.Y.A.

33. Some references: [1] K. Stefański: Chaotic transients in multidimenional maps, Rep. Math. Phys. 44, 231-240 (1999). [2] K. Stefański, K. Buszko, K. Piecyk: Transient chaosmeasurements using finite-timeLyapunov exponents, Chaos 20, 033117-033117-13 (2010). [3] K.Stefański: Quantum description of a dye laser thresholdregion, Z. Phys. B 45, 351-361 (1982).