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

2. Plan Motivation Weyl Wigner formalism Quadratic case ( exact ) General case ( semiclassical ) Conclusion

3. Motivations How to separate a wavy behaviour from a ballistic one in experimental data? When does it become impossible / possible to describe the results classically?

4. 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!

5. Wigner function How does it look like?

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

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

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

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

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

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

12. Decoherence time Elliptic case Hyperbolic case

13. Semiclassical generalization

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

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

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

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

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

19. Hamilton Jacobi equation Centre→Chord propagator

20. Small chords limit

21. With environment (non unitary) In the small chords limit: Liouville PropagationGaussian cut outAiry function

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

23. Results

24. Conclusion Quadratic case: transition from a quantum regime to a purely classic one ( positivity threshold ). Exactly solvable. General case: transition as well. Decoherence is position dependent. No analytical solution but numerically accessible results (classical runge kutta).