Semiclassical Propagator of the Wigner Function

Dittrich, Thomas; Viviescas, Carlos; Sandoval, Luis

doi:10.1103/physrevlett.96.070403

Cited by 51 publications

(76 citation statements)

References 19 publications

(20 reference statements)

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“…At first sight, this centre-centre propagator seems to result from a double Weyl transform from the single Weyl propagator, U(x), of the unitary mapÛ (see [27,28]), but the derivation of (56) clarifies its true role as the inverse transform from the double Wigner function, that is the Choi representation of the super evolution operator. ¶ In the semiclassical regime the ordinary Weyl propagators have explicit formulae in terms of generating functions [31,16].…”

Section: Quantum Evolutionsmentioning

confidence: 99%

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Representation of superoperators in double phase space

Saraceno

Almeida

2016

J. Phys. A: Math. Theor.

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Abstract. Operators in quantum mechanics -either observables, density or evolution operators, unitary or not -can be represented by c-numbers in operator bases. The position and momentum bases are in one to one correspondence with lagrangian planes in double phase space, but this is also true for the well known Wigner-Weyl correspondence based on translation and reflection operators. These phase space methods are here extended to the representation of superoperators. We show that the Choi-Jamiolkowsky isomorphism between the dynamical matrix and the linear action of the superoperator constitutes a "double" Wigner or chord transform when represented in double phase space. As a byproduct several previously unknown integral relationships between products of Wigner and chord distributions for pure states are derived.

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Section: Quantum Evolutionsmentioning

confidence: 99%

“…that is, both correlations can then be computed directly from the chord and the (ordinary) Wigner function and chord function (28). In the case of pure states,ρ = |ψ ψ|, the correlations can be computed directly:…”

Section: Phase Space Correlations and General Pure State Conditionsmentioning

confidence: 99%

Representation of superoperators in double phase space

Saraceno

Almeida

2016

J. Phys. A: Math. Theor.

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“…It facilitates developping and assessing semiclassical approximations for the propagator in such systems [10], providing an alternative point of view for the analysis of coherent structures like "quantum carpets" [11,12].…”

Section: Resultsmentioning

confidence: 99%

Wigner function for discrete phase space: Exorcising ghost images

Argüelles

Dittrich

2005

Physica A: Statistical Mechanics and its Applications

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We construct, using simple geometrical arguments, a Wigner function defined on a discrete phase space of arbitrary integer Hilbert-space dimension that is free of redundancies. "Ghost images" plaguing other Wigner functions for discrete phase spaces are thus revealed as artifacts. It allows to devise a corresponding phase-space propagator in an unambiguous manner. We apply our definitions to eigenstates and propagator of the quantum baker map. Scars on unstable periodic points of the corresponding classical map become visible with unprecedented resolution.

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“…This has the peculiarity that the dominant classical trajectory is precisely on a caustic, so that a higher uniform approximation has been developed, for which the phases were previously analyzed [20,22]. In contrast, the mixed propagator presented in [21] avoids the caustic, so that its tangent propagator for short times is given by (3.2) with a positive sign.…”

Section: Products Of Metaplectic Transformationsmentioning

confidence: 99%

Metaplectic sheets and caustic traversals in the Weyl representation

Almeida

Ingold

2014

J. Phys. A: Math. Theor.

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Abstract. The quantum Hamiltonian generates in time a family of evolution operators. Continuity of this family holds within any choice of representation and, in particular, for the Weyl propagator, even though its simplest semiclassical approximation may develop caustic singularities. The phase jumps of the Weyl propagator across caustics have not been previously determined.The semiclassical appproximation relies on individual classical trajectories together with their neighbouring tangent map. Based on the latter, one defines a continuous family of unitary tangent propagators, with an exact Weyl representation that is close to the full semiclassical approximation in an appropriate neighbourhood. The phase increment of the semiclassical Weyl propagator, as a caustic is crossed, is derived from the facts that the corresponding family of tangent operators belong to the metaplectic group and that the products of the tangent propagators are obtained from Gaussian integrals. The Weyl representation of the metaplectic group is here presented, with the correct phases determined within an intrinsic ambiguity for the overall sign. The elements that fully determine the phase increment across a particular caustic are then analysed.

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Semiclassical Propagator of the Wigner Function

Cited by 51 publications

References 19 publications

Representation of superoperators in double phase space

Representation of superoperators in double phase space

Wigner function for discrete phase space: Exorcising ghost images

Metaplectic sheets and caustic traversals in the Weyl representation

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