Decoherence in Phase Space for Markovian Quantum Open Systems

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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
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.

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Separation time
Breakdown of correspondence in chaotic systems
Ehrenfest versus localization times Zbyszek P.
Karkuszewski, Jakub Zakrzewski, Wojciech H.
Zurek Phys. Rev. A 65, 042113 (2002)
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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)
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Separation time and decoherence
Decoherence, Chaos, and the Correspondence
PrincipleSalman Habib, Kosuke Shizume, Wojciech
Hubert ZurekPhys.Rev.Lett. 80 (1998)
4361-4365 
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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?
p
p
q
q
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Fourier Transform
Wigner function W(x) ? Chord function ?(?)
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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Which System?
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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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1 - simple case quadratic system
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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
a is a parameter related to the coupling strength
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Decoherence time / dynamics
a0.001
Elliptic case
Log
a1
Hyperbolic case
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2 - semiclassical generalizationa - without
environment
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W.K.B.
Approximate solution of the Schrödinger equation
Hamilton-Jacobi
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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
Centre?Centre propagator
Chord space
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
Stationnary phase
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Small chords limit
?
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b - with environnement
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With environment (non unitary)
In the small chords limit
Airy function
Liouville Propagation
Gaussian cut out

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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 To be continued
• Decoherence is not uniform in phase space.
• No analytical solution but numerically
  accessible results (classical runge kutta).