Symplectic evolution of Wigner functions in markovian open systems

Abstract

The Wigner function is known to evolve classically under the exclusive action of a quadratic hamiltonian. If the system does interact with the environment through Lindblad operators that are linear functions of position and momentum, we show that the general evolution is the convolution of the classically evolving Wigner function with a phase space gaussian that broadens in time. We analyze the three generic cases of elliptic, hyperbolic and parabolic Hamiltonians. The Wigner function always becomes positive in a definite time, which is shortest in the hyperbolic case. We also derive an exact formula for the evolving linear entropy as the average of a narrowing gaussian taken over a probability distribution that depends only on the initial state. This leads to a long time asymptotic formula for the growth of linear entropy.

this new version treats the dissipative case

References in corpus (5)

• Open quantum systems
• Quantum chaos: a decoherent definition.
• Exact positivity of the Wigner and P-functions of a Markovian open system
• Recovering coherence from decoherence: a method of quantum state reconstruction
• Decoherence of Semiclassical Wigner Functions

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