1 / 29

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

emma
Download Presentation

Decoherence in Phase Space for Markovian Quantum Open Systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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).

More Related