How to calculate the Wigner function from phase space

Hug, M.; Menke, Christoph; Schleich, Wolfgang P.

doi:10.1088/0305-4470/31/11/002

Cited by 11 publications

(5 citation statements)

References 9 publications

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“…As a nontrivial illustration, the Wigner functions, W , associated with several eigenstates of the Morse potential, have been studied by combining analytical and numerical methods: negative values of the W associated with the ground state are reported in [42,43]. The latter two references and [44] present negative values of the W 's associated with some excited states. Even if W st < 0, the domain in which that occurs cannot be large and has to be consistent with the fact that both…”

Section: W and W St As Quasi-definite Functionals In Momentummentioning

confidence: 99%

Non-Equilibrium Liouville and Wigner Equations: Moment Methods and Long-Time Approximations

Alvarez‐Estrada

2014

Entropy

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Abstract:We treat the non-equilibrium evolution of an open one-particle statistical system, subject to a potential and to an external "heat bath" (hb) with negligible dissipation. For the classical equilibrium Boltzmann distribution, W c,eq , a non-equilibrium three-term hierarchy for moments fulfills Hermiticity, which allows one to justify an approximate long-time thermalization. That gives partial dynamical support to Boltzmann's W c,eq , out of the set of classical stationary distributions, W c,st , also investigated here, for which neither Hermiticity nor that thermalization hold, in general. For closed classical many-particle systems without hb (by using W c,eq ), the long-time approximate thermalization for three-term hierarchies is justified and yields an approximate Lyapunov function and an arrow of time. The largest part of the work treats an open quantum one-particle system through the non-equilibrium Wigner function, W . W eq for a repulsive finite square well is reported. W 's (< 0 in various cases) are assumed to be quasi-definite functionals regarding their dependences on momentum (q). That yields orthogonal polynomials, H Q,n (q), for W eq (and for stationary W st ), non-equilibrium moments, W n , of W and hierarchies. For the first excited state of the harmonic oscillator, its stationary W st is a quasi-definite functional, and the orthogonal polynomials and three-term hierarchy are studied. In general, the non-equilibrium quantum hierarchies (associated with W eq ) for the W n 's are not three-term ones. As an illustration, we outline a non-equilibrium four-term hierarchy and its solution in terms of generalized operator continued fractions. Such structures also allow one to formulate long-time approximations, but make it more difficult to justify thermalization. For large thermal and de Broglie wavelengths, the dominant W eq and a non-equilibrium equation for W are reported: the non-equilibrium hierarchy could plausibly be a three-term one and possibly not far from Gaussian, and thermalization could possibly be justified. Liouville and Wigner distributions and non-equilibrium equations; equilibrium solutions and orthogonal polynomials; non-equilibrium moments; long-time approximations; low temperature regime

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Section: W and W St As Quasi-definite Functionals In Momentummentioning

confidence: 99%

Non-Equilibrium Liouville and Wigner Equations: Moment Methods and Long-Time Approximations

Alvarez‐Estrada

2014

Entropy

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“…We choose the Morse oscillator for a detailed study of semiclassical Wigner propagation for two reasons: Above all, being prototypical for strongly anharmonic molecular potentials and correspondingly complex dynamics [67][68][69][70], it is a widely used benchmark for numerical methods in this realm [4,21,[71][72][73]. Secondly, the Morse oscillator has the remarkable advantage, shared only with very few other potentials, that closed analytical expressions are available not only for its energy eigenstates [74] but even for their Wigner representations [71], a feature that greatly facilitates the comparison with exact quantum-mechanical results [43,71,[75][76][77][78] and hence provides a solid basis for an objective test of the performance of semiclassical approximations to the propagator.…”

Section: Morse Oscillatormentioning

confidence: 99%

Semiclassical propagation of Wigner functions

Dittrich,

Gomez,

Pachon

2009

Preprint

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We present a comprehensive study of semiclassical phase-space propagation in the Wigner representation, emphasizing numerical applications, in particular as an initial-value representation.Two semiclassical approximation schemes are discussed: The propagator of the Wigner function based on van Vleck's approximation replaces the Liouville propagator by a quantum spot with an oscillatory pattern reflecting the interference between pairs of classical trajectories. Employing phase-space path integration instead, caustics in the quantum spot are resolved in terms of Airy functions. We apply both to two benchmark models of nonlinear molecular potentials, the Morse oscillator and the quartic double well, to test them in standard tasks such as computing autocorrelation functions and propagating coherent states. The performance of semiclassical Wigner propagation is very good even in the presence of marked quantum effects, e.g., in coherent tunneling and in propagating Schrödinger cat states, and of classical chaos in four-dimensional phase space.We suggest options for an effective numerical implementation of our method and for integrating it in Monte-Carlo-Metropolis algorithms suitable for high-dimensional systems.

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“…In the resulting one-index recurrences the coefficients will be matrices acting on appropriate vectors formed with the C α n . ‡ Dynamical equations for the Wigner function have anyway been tackled by purely numerical methods, like (pseudo-) spectral methods for partial differential equations (see [42,43] and references therein), or grid discretization usually followed by Runge-Kutta integration [44].…”

Section: Matrix Quantum Hierarchies With One-index Recurrencesmentioning

confidence: 99%

The Caldeira–Leggett quantum master equation in Wigner phase space: continued-fraction solution and application to Brownian motion in periodic potentials

García‐Palacios¹,

Zueco²

2004

J. Phys. A: Math. Gen.

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The continued-fraction method to solve classical Fokker-Planck equations has been adapted to tackle quantum master equations of the Caldeira-Leggett type. This can be done taking advantage of the phase-space (Wigner) representation of the quantum density matrix. The approach differs from those in which some continued-fraction expression is found for a certain quantity, in that the full solution of the master equation is obtained by continued-fraction methods. This allows to study in detail the effects of the environment (fluctuations and dissipation) on several classes of nonlinear quantum systems. We apply the method to the canonical problem of quantum Brownian motion in periodic potentials both for cosine and ratchet potentials (lacking inversion symmetry).

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How to calculate the Wigner function from phase space

Cited by 11 publications

References 9 publications

Non-Equilibrium Liouville and Wigner Equations: Moment Methods and Long-Time Approximations

Non-Equilibrium Liouville and Wigner Equations: Moment Methods and Long-Time Approximations

Semiclassical propagation of Wigner functions

The Caldeira–Leggett quantum master equation in Wigner phase space: continued-fraction solution and application to Brownian motion in periodic potentials

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