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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
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Plan Motivation Weyl Wigner formalism Quadratic case ( exact ) General case ( semiclassical ) Conclusion
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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?
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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!
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Wigner function How does it look like?
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Fourier Transform Wigner function W(x) → Chord function χ(ξ) Semiclassical origin of “chord” dubbing: Centre → Chord
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Physical analogy Small chords → Classical features ( direct transmission ) Large chords → Quantum fringes ( lateral repetition pattern )
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Markovian Quantum Open System General form for the time evolution of a reduced density operator : Lindblad equation. Reduced Density Operator:
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Quadratic Hamiltonian with linear coupling to environment: Weyl representation Centre space: Fockker-Planck equation Chord space:
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Behaviour of the solution The Wigner function is: - Classically propagated - Coarse grained It becomes positive
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Analytical expression The chord function is cut out The Wigner function is coarse grained With: α is a parameter related to the coupling strength
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Decoherence time Elliptic case Hyperbolic case
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Semiclassical generalization
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W.K.B. Hamilton-Jacobi: Approximate solution of the Schrödinger equation:
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W.K.B. in Doubled Phase Space
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Propagator for the Wigner function (Unitary case) Reflection Operator: Time evolution:
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Weyl representation of the propagator Centre space: Chord space: Centre→Centre propagator Centre→Chord propagator
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WKB ansatz The Centre→Chord propagator is initially caustic free We infer a WKB anstaz for later time:
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Hamilton Jacobi equation Centre→Chord propagator
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Small chords limit
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With environment (non unitary) In the small chords limit: Liouville PropagationGaussian cut outAiry function
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Application to moments Justifies the small chords approximation For instance:
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Results
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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).
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