dorsal/arxiv
View SchemaDecoherence of Semiclassical Wigner Functions
| Authors | Alfredo M. Ozorio de Almeida |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0208094 |
| URL | https://arxiv.org/abs/quant-ph/0208094 |
| DOI | 10.1088/0305-4470/36/1/305 |
Abstract
The Lindblad equation governs general markovian evolution of the density operator in an open quantum system. An expression for the rate of change of the Wigner function as a sum of integrals is one of the forms of the Weyl representation for this equation. The semiclassical description of the Wigner function in terms of chords, each with its classically defined amplitude and phase, is thus inserted in the integrals, which leads to an explicit differential equation for the Wigner function. All the Lindblad operators are assumed to be represented by smooth phase space functions corresponding to classical variables. In the case that these are real, representing hermitian operators, the semiclassical Lindblad equation can be integrated. There results a simple extension of the unitary evolution of the semiclassical Wigner function, which does not affect the phase of each chord contribution, while dampening its amplitude. This decreases exponentially, as governed by the time integral of the square difference of the Lindblad functions along the classical trajectories of both tips of each chord. The decay of the amplitudes is shown to imply diffusion in energy for initial states that are nearly pure. Projecting the Wigner function onto an orthogonal position or momentum basis, the dampening of long chords emerges as the exponential decay of off-diagonal elements of the density matrix.
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"abstract": "The Lindblad equation governs general markovian evolution of the density\noperator in an open quantum system. An expression for the rate of change of the\nWigner function as a sum of integrals is one of the forms of the Weyl\nrepresentation for this equation. The semiclassical description of the Wigner\nfunction in terms of chords, each with its classically defined amplitude and\nphase, is thus inserted in the integrals, which leads to an explicit\ndifferential equation for the Wigner function. All the Lindblad operators are\nassumed to be represented by smooth phase space functions corresponding to\nclassical variables. In the case that these are real, representing hermitian\noperators, the semiclassical Lindblad equation can be integrated. There results\na simple extension of the unitary evolution of the semiclassical Wigner\nfunction, which does not affect the phase of each chord contribution, while\ndampening its amplitude. This decreases exponentially, as governed by the time\nintegral of the square difference of the Lindblad functions along the classical\ntrajectories of both tips of each chord. The decay of the amplitudes is shown\nto imply diffusion in energy for initial states that are nearly pure.\nProjecting the Wigner function onto an orthogonal position or momentum basis,\nthe dampening of long chords emerges as the exponential decay of off-diagonal\nelements of the density matrix.",
"arxiv_id": "quant-ph/0208094",
"authors": [
"Alfredo M. Ozorio de Almeida"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/36/1/305",
"title": "Decoherence of Semiclassical Wigner Functions",
"url": "https://arxiv.org/abs/quant-ph/0208094"
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