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Decoherence in Phase Space for Markovian Quantum Open Systems

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. Plan. Motivation: quantum-classical correspondence Weyl Wigner formalism: mapping quantum onto classical

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Decoherence in Phase Space for Markovian Quantum Open Systems

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  1. Decoherence in Phase Spacefor 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 timesZbyszek P. Karkuszewski, Jakub Zakrzewski, Wojciech H. Zurek Phys. Rev. A 65, 042113 (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, 026211 (2004)

  5. Separation time and decoherence Decoherence, Chaos, and the Correspondence PrincipleSalman Habib, Kosuke Shizume, Wojciech Hubert ZurekPhys.Rev.Lett. 80 (1998) 4361-4365

  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 generalizationa - 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: Airy function Liouville Propagation Gaussian cut out …

  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 solutionbut numerically accessible results (classical runge kutta).

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