Moyal Equation Research Articles

The Moyal equation for open quantum systems

We generalize the Moyal equation, which describes the dynamics of quantum observables in phase space, to quantum systems coupled to a reservoir. It is shown that phase space observables become functionals of fluctuating noise forces introduced by the coupling to the reservoir. For Markovian reservoirs, the Moyal equation turns into a functional differential equation in which the reservoir’s effect can be described by a single parameter.

Nonlinear evolution of a single coherent mode in a turbulent plasma

We consider the excitation, nonlinear evolution and saturation of a single coherent mode in a turbulent plasma background. We adopt a generic wave-kinetic description of plasma turbulence, based on a Wigner–Moyal equation, which stays valid well beyond the geometric optics approximation. We illustrate our model by considering the case of a geodesic acoustic mode in a tokamak plasma, in the presence of a broad spectrum of drift wave turbulence.

Wigner function and the probability representation of quantum states

The relation of theWigner function with the fair probability distribution called tomographic distribution or quantum tomogram associated with the quantum state is reviewed. The connection of the tomographic picture of quantum mechanics with the integral Radon transform of the Wigner quasidistribution is discussed. The Wigner–Moyal equation for the Wigner function is presented in the form of kinetic equation for the tomographic probability distribution both in quantum mechanics and in the classical limit of the Liouville equation. The calculation of moments of physical observables in terms of integrals with the state tomographic probability distributions is constructed having a standard form of averaging in the probability theory. New uncertainty relations for the position and momentum are written in terms of optical tomograms suitable for directexperimental check. Some recent experiments on checking the uncertainty relations including the entropic uncertainty relations are discussed.

Diffusion equations and the time evolution of foreign exchange rates

Abstract We investigate which type of diffusion equation is most appropriate to describe the time evolution of foreign exchange rates. We modify the geometric diffusion model assuming a non-exponential time evolution and the stochastic term is the sum of a Wiener noise and a jump process. We find the resulting diffusion equation to obey the Kramers–Moyal equation. Analytical solutions are obtained using the characteristic function formalism and compared with empirical data. The analysis focus on the first four central moments considering the returns of foreign exchange rate. It is shown that the proposed model offers a good improvement over the classical geometric diffusion model.

Convergence of a quantum normal form and an exact quantization formula

The operator −iℏω⋅∇ on L2(Tl), quantizing the linear flow of diophantine frequencies ω=(ω1,…,ωl) over Tl, l>1, is perturbed by the operator quantizing a function Vω(ξ,x)=V(ω⋅ξ,x):Rl×Tl→R, z↦V(z,x):R×Tl→R real-holomorphic. The corresponding quantum normal form (QNF) is proved to converge uniformly in ℏ∈[0,1]. This yields non-trivial examples of quantum integrable systems, an exact quantization formula for the spectrum, and a convergence criterion for the Birkhoff normal form, valid for perturbations holomorphic away from the origin. The main technical aspect concerns the solution of the quantum homological equation, which is constructed and estimated by solving the Moyal equation for the operator symbols. The KAM iteration can thus be implemented on the symbols, and its convergence proved. This entails the convergence of the QNF, with radius estimated in terms only of the diophantine constants of ω.

Probability representation of the quantum evolution and energy-level equations for optical tomograms

The von Neumann evolution equation for the density matrix and the Moyal equation for the Wigner function are mapped onto the evolution equation for the optical tomogram of the quantum state. The connection with the known evolution equation for the symplectic tomogram of the quantum state is clarified. The stationary states corresponding to quantum energy levels are associated with the probability representation of the von Neumann and Moyal equations written for optical tomograms. The classical Liouville equation for optical tomogram is obtained. An example of the parametric oscillator is considered in detail.

Moyal and tomographic probability representations for f-oscillator quantum states

States of nonlinear quantum oscillators (f-oscillators) are considered in the Weyl–Wigner–Moyal representation and the tomographic probability representation, where the states are described by standard probability distributions instead of wave functions or density matrices. The evolving integrals of motion for classical and quantum f-oscillators are found and the solution for the Liouville equation associated with the probability distribution on the phase space for this oscillator is obtained along with the solution of the Moyal equation for the quantum f-oscillator, which provide solutions for the partial case of f-nonlinearity existing in Kerr media. Nonlinear coherent states and the thermodynamics of nonlinear oscillators are studied.

Moyal phase-space analysis of nonlinear optical Kerr media

Nonlinear optical media of Kerr type are described by a particular version of an anharmonic quantum harmonic oscillator. The dynamics of this system can be described using the Moyal equations of motion, which correspond to a quantum phase-space representation of the Heisenberg equations of motion. For the Kerr system we derive exact solutions of the Moyal equations for an irreducible set of observables. These Moyal solutions incorporate the asymptotics of the classical limit in a simple, explicit form. An unusual feature of these solutions is that they exhibit periodic singularities in the time variable. These singularities are removed by the phase-space averaging required to construct the expectation value for an arbitrary initial state. Nevertheless, for strongly number-squeezed initial states the effects of the singularity remain observable.

Mutual coherence of polarized light in disordered media: two-frequency method extended

The paper addresses the two-point correlations of electromagnetic waves in general random, bi-anisotropic media whose constitutive tensors are complex Hermitian, positive- or negative-definite matrices. A simplified version of the two-frequency Wigner distribution (2f-WD) for polarized waves is introduced and the closed form Wigner–Moyal equation is derived from the Maxwell equations. In the weak-disorder regime with an arbitrarily varying background the two-frequency radiative transfer (2f-RT) equations for the associated 2 × 2 coherence matrices are derived from the Wigner–Moyal equation by using the multiple-scale expansion. In birefringent media, the coherence matrix becomes a scalar and the 2f-RT equations take the scalar form due to the absence of depolarization. A paraxial approximation is developed for spatially anisotropic media. Examples of isotropic, chiral, uniaxial and gyrotropic media are discussed.

Influence of nonlinear dispersion on modulation instability of coherent and partially coherent ultrashort pulses in metamaterials

The combination of dispersive magnetic permeability with nonlinear polarization leads to a series of nonlinear dispersion terms in the propagation equations for ultrashort pulses in metamaterials. Here we present an investigation of modulation instability (MI) of both coherent and partially coherent ultrashort pulses in metamaterials to identify the role of nonlinear dispersion in pulse propagation. The Wigner–Moyal equation for partially coherent ultrashort pulses and the nonlinear dispersion relation for MI in metamaterials are derived. Combining the standard MI theory with the unique properties of the metamaterial, the influence of the controllable first-order nonlinear dispersion, namely self-steepening, and the second-order nonlinear dispersion on both coherent and partially coherent MI, in both negative-index and positive-index regions of the metamaterial for all physically possible cases is analyzed in detail. For the first time to our knowledge, we demonstrate that the role of the second-order nonlinear dispersion in MI is equivalent to that of group-velocity dispersion (GVD) to some extent, and thus due to the role of the second-order nonlinear dispersion, MI may appear in the otherwise impossible cases, such as in the normal GVD regime.

Features of Moyal trajectories

We study the Moyal evolution of the canonical position and momentum variables. We compare it with the classical evolution and show that, contrary to what is commonly found in the literature, the two dynamics do not coincide. We prove that this divergence is quite general by studying Hamiltonians of the form p2∕2m+V(q). Several alternative formulations of Moyal dynamics are then suggested. We introduce the concept of star function and use it to reformulate the Moyal equations in terms of a system of ordinary differential equations on the noncommutative Moyal plane. We then use this formulation to study the semiclassical expansion of Moyal trajectories, which is cast in terms of a (order by order in ℏ) recursive hierarchy of (i) first order partial differential equations as well as (ii) systems of first order ordinary differential equations. The latter formulation is derived independently for analytic Hamiltonians as well as for the more general case of locally integrable ones. We present various examples illustrating these results.

Local and non-local force analysis for Wigner function barrier scattering

The Derivative Propagation Method (DPM) can be used to classify the components of the total force acting upon a trajectory into local and non-local components, without the use of finite differences or function fittings. In this study, the DPM was used to generate and analyze classical local and quantum non-local forces for evolving trajectories given by a phase space Wigner distribution function governed by the Wigner–Moyal equation for Eckart barrier scattering. Solutions were verified with finite difference methods.

White-Noise and Geometrical OpticsLimits of Wigner–Moyal Equation for Beam Waves in Turbulent Media II: Two-Frequency Formulation

We introduce two-frequency Wigner distribution in the setting of parabolic approximation to study the scaling limits of the wave propagation in a turbulent medium at two different frequencies. We show that the two-frequency Wigner distribution satisfies a closed-form equation (the two-frequency Wigner–Moyal equation). In the white-noise limit we show the convergence of weak solutions of the two-frequency Wigner–Moyal equation to a Markovian model and thus prove rigorously the Markovian approximation with power-spectral densities widely used in the physics literature. We also prove the convergence of the simultaneous geometrical optics limit whose mean field equation has a simple, universal form and is exactly solvable.

Wave-kinetic description of nonlinear photons

The nonlinear interaction, due to quantum electrodynamical effects, between photons is investigated using a wave-kinetic description. Starting from a coherent wave description, we use the Wigner transform technique to obtain a set of wave-kinetic equations, the so called Wigner–Moyal equations. These equations are coupled to a background radiation fluid, whose dynamics are determined by an acoustic wave equation. In the slowly varying acoustic limit, we analyse the resulting system of kinetic equations, and show that they describe instabilities, as well as Landau-like damping. The instabilities may lead to the break-up and focusing of ultra-high-intensity multi-beam systems, which in conjunction with the damping may result in stationary strong field structures. The results could be of relevance for the next generation of laser–plasma systems.

Propagators in the Moyal and Tomographic Representations of States

The tomographic map and density operator description of quantum states are reviewed. The connection between the tomogram and the probability distribution function is discussed. The relation of the Green function of the Moyal equation for the Wigner function to the Green function of the Schrodinger (von Neumann) equation in the tomographic representation is studied. An example of a quantum system with a quadratic Hamiltonian is considered.

Nonlinear Dynamics of Partially Incoherent Optical Waves Based on the Wigner Transform Method

A novel statistical approach based on the Wigner transform method is proposed for the description of partially incoherent optical wave dynamics in nonlinear media. The Wigner–Moyal equation is derived for the Wigner distribution of the optical wave field governed by the nonlinear Schrödinger equation with an arbitrary nonlinearity. An application to incoherent light propagation in dispersive Kerr media shows that random phase fluctuations of a plane wave solution lead to a linear Landau-like damping effect, which can stabilize the nonlinear modulational instability. A similar effect is shown to occur in the case of the two-stream instability of two partially incoherent optical waves interacting with each other through cross-phase modulation initiated by the nonlinearity. In the limit of the geometrical optics approximation, it is shown that 1D and 2D self-trapped, stationary and incoherent wave pulse structures may exist for a wide class of nonlinear media. Furthermore, time-dependent self-similar 1D and 2D structures have been found in the case of the Kerr nonlinearity.

New results on Fokker–Planck equations of fractional order

It is shown that the fractional Fokker–Planck equations proposed recently in the literature (by merely substituting time fractional derivative for time derivative) give rise to some problems in the sense that they provide probability densities which may have negative values. In the same way, one shows that the Kramers–Moyal equation can be thought of as related to fractal processes, but it is well known that it yields also negative densities. It seems that the key of this trouble is the misuse of the Chapman Kolmogorov equation on the one hand, and of the fractional difference on the other hand. In fact, there is a complete identification between Kramers–Moyal equation and Fokker–Planck equation of fractional order. After a careful analysis, one arrives at the conclusion that the fractional derivative in Liouville–Riemann (L–R) sense should be replaced by a slightly finite fractional derivative which involves finite difference, whilst L–R fractional derivative refers to difference of infinite order. The new fractional Fokker–Planck equation so obtained is displayed, and its solution via separation of variables is outlined. It seems that there is no alternative but to work via non-standard analysis, that is to say infinitesimal discretization in time.

Moyal Quantum Mechanics: The Semiclassical Heisenberg Dynamics

The Moyal description of quantum mechanics, based on the Wigner-Weyl isomorphism between operators and symbols, provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in ħ and so presents a preferred foundation for semiclassical analysis. Its semiclassical expansion "coefficients," acting on symbols that represent observables, are simple, globally defined (phase space) differential operators constructed in terms of the classical flow. The first of two presented methods introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold′s formula for the Weyl product of two symbols and has ħ as its natural small parameter. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of "quantum trajectories." Their Green function solutions construct the regular ħ ⇓ 0 asymptotic series for the Heisenberg-Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the ħ coefficients recursively. In contrast to the WKB approximation for propagators, the Heisenberg-Weyl description of evolution involves no essential singularity in ħ, no Hamilton-Jacobi equation to solve for the action, and no multiple trajectories, caustics, or Maslov indices.

Erratum: Solution of Kramers–Moyal equations for problems in chemical physics [J. Chem. Phys. 8 1, 1285 (1984)

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Erratum: Solution of Kramers–Moyal equations for problems in chemical physics [J. Chem. Phys. 81, 1285 (1984)

Erratum: Solution of Kramers–Moyal equations for problems in chemical physics [J. Chem. Phys. <b>81</b>, 1285 (1984)