Download Decoherence in phase space for Markovian Quantum Open Systems

Decoherence in Phase Space
for 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.
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)
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)
Separation time and decoherence
Decoherence, Chaos, and the Correspondence
Principle
Salman Habib, Kosuke Shizume, Wojciech
Hubert Zurek
Phys.Rev.Lett. 80 (1998) 4361-4365
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 )
Which System?
Markovian Quantum Open
System
General form for the time
evolution of a reduced density
operator : Lindblad equation.
Reduced Density Operator:
1 - simple case: quadratic system
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
Log
Elliptic case
α=1
Hyperbolic case
2 - semiclassical generalization
a - without environment
W.K.B.
Approximate solution of the
Schrödinger equation:
Hamilton-Jacobi:
W.K.B. in Doubled Phase Space
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
Small chords limit
ξ
b - with environnement
With environment (non unitary)
In the small chords limit:
Liouville Propagation
Gaussian cut out
Airy function
…
Application to moments
Justifies the small chords approximation
For instance:
Results
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).