1. Decoherence in Phase Space for Markovian Quantum Open Systems
Olivier Brodier1
& Alfredo M. Ozorio de Almeida2
1 – M.P.I.P.K.S. Dresden
2 – C.B.P.F. Rio de Janeiro

2. Plan Motivation: quantum-classical correspondence
Weyl Wigner formalism: mapping quantum onto classical
Markovian open quantum system, quadratic case: exact classical analogy
General case: a semiclassical approach
Conclusion: analytically accessible or numerically cheap.

3. Separation time
Breakdown of correspondence in chaotic systems: Ehrenfest versus
localization times Zbyszek P. Karkuszewski, Jakub Zakrzewski, Wojciech H. Zurek Phys. Rev. A 65, (2002)

4. Separation time
Environmental effects in the quantum-classical transition for the delta-kicked
harmonic oscillator
A.R.R. Carvalho, R. L. de Matos Filho, L. Davidovich Phys. Rev. E 70, (2004)

5. Separation time and decoherence
Decoherence, Chaos, and the Correspondence Principle Salman Habib, Kosuke Shizume, Wojciech Hubert Zurek Phys.Rev.Lett. 80 (1998)  

6. Weyl Representation
To map the quantum problem onto a classical frame: the phase space.
Analogous to a classical probability distribution in phase space.
BUT: W(x) can be negative!

7. Wigner function
How does it look like? p p q q

8. Fourier Transform Wigner function W(x) → Chord function χ(ξ)
Semiclassical origin
of “chord” dubbing:
Centre → Chord

9. Physical analogy
Small chords → Classical features ( direct transmission )
Large chords → Quantum fringes ( lateral repetition pattern )

10. Which System?

11. Markovian Quantum Open System
General form for the time evolution of a reduced density operator : Lindblad equation.
Reduced Density Operator:

12. 1 - simple case: quadratic system

13. Quadratic Hamiltonian with linear coupling to environment: Weyl representation
Centre space: Fockker-Planck equation
Chord space:

14. Behaviour of the solution
The Wigner function is:
Classically propagated - Coarse grained
It becomes positive

15. Analytical expression
The chord function is cut out
The Wigner function is coarse grained
With:
α is a parameter related to the coupling strength

16. Decoherence time / dynamics
α=0.001
Elliptic case
Log
α=1
Hyperbolic case

17. 2 - semiclassical generalization a - without environment

18. W.K.B. Approximate solution of the Schrödinger equation:
Hamilton-Jacobi:

19. W.K.B. in Doubled Phase Space

20. Propagator for the Wigner function (Unitary case)
Reflection Operator:
Time evolution:

21. Weyl representation of the propagator
Centre space:
Centre→Centre propagator
Chord space:
Centre→Chord propagator

22. WKB ansatz The Centre→Chord propagator is initially caustic free
We infer a WKB anstaz for later time:

23. Hamilton Jacobi equation
Centre→Chord propagator
Stationnary phase

24. Small chords limit ξ

25. b - with environnement

26. With environment (non unitary)
In the small chords limit:
Liouville Propagation
Gaussian cut out
Airy function …

27. Application to moments
Justifies the small chords approximation
For instance:

28. Results

29. Conclusion
Quadratic case: transition from a quantum regime to a purely classic one ( positivity threshold ). Exactly solvable.
General case: To be continued…
Decoherence is not uniform in phase space.
No analytical solution but numerically accessible results (classical runge kutta).