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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. Plan. Motivation: quantum-classical correspondence Weyl Wigner formalism: mapping quantum onto classical
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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
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.
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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!
Wigner function How does it look like? p p q q
Fourier Transform Wigner function W(x) → Chord function χ(ξ) Semiclassical origin of “chord” dubbing: Centre → Chord
Physical analogy Small chords → Classical features ( direct transmission ) Large chords → Quantum fringes ( lateral repetition pattern )
Markovian Quantum Open System General form for the time evolution of a reduced density operator : Lindblad equation. Reduced Density Operator:
Quadratic Hamiltonian with linear coupling to environment: Weyl representation Centre space: Fockker-Planck equation Chord space:
Behaviour of the solution • The Wigner function is: • Classically propagated- Coarse grained • It becomes positive
Analytical expression The chord function is cut out The Wigner function is coarse grained With: α is a parameter related to the coupling strength
Decoherence time / dynamics α=0.001 Elliptic case Log α=1 Hyperbolic case
W.K.B. Approximate solution of the Schrödinger equation: Hamilton-Jacobi:
Propagator for the Wigner function(Unitary case) Reflection Operator: Time evolution:
Weyl representation of the propagator Centre space: Centre→Centre propagator Chord space: Centre→Chord propagator
WKB ansatz The Centre→Chord propagator is initially caustic free We infer a WKB anstaz for later time:
Hamilton Jacobi equation Centre→Chord propagator Stationnary phase
With environment (non unitary) In the small chords limit: Airy function Liouville Propagation Gaussian cut out …
Application to moments Justifies the small chords approximation For instance:
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).