Decoherence in Phase Spacefor Markovian Quantum Open Systems

Olivier Brodier1

& Alfredo M. Ozorio de Almeida2

1 – M.P.I.P.K.S. Dresden2 – C.B.P.F. Rio de Janeiro

Plan

• Motivation

• Weyl Wigner formalism

• Quadratic case ( exact )

• General case ( semiclassical )

• Conclusion

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?

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?

Fourier Transform

Wigner function W(x) → Chord function χ(ξ)

Semiclassical originof “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 expressionThe chord function is cut out

The Wigner function is coarse grained

With:

α is a parameter related to the coupling strength

Decoherence time

Elliptic case

Hyperbolic case

Semiclassical generalization

W.K.B.

Hamilton-Jacobi:

Approximate solution of the Schrödinger equation:

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:

Chord space:

Centre→Centre propagator

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

Small chords limit

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: transition as well. Decoherence is position dependent.

No analytical solution but numerically accessible results (classical runge kutta).