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QUANTUM METHODOLOGIES IN BEAM, FLUID AND PLASMA PHYSICS

Renato Fedele

Published Mar 1, 2008• 0 citations
Publisher

Abstract

The quantum methodologies useful for describing in a unified way several problems of nonlinear and collective dynamics of fluids, plasmas and beams are presented. In particular, the pictures given by the Madelung fluid and the Moyal-Ville-Wigner phase-space quasidistribution, including the related quantum tools such as marginal distributions for the tomographic representations, are described. Some relevant applications to soliton and modulational instability theory are presented.

DOI
10.1142/9789812799784_0005

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The quantum methodologies useful for describing in a unified way several problems of nonlinear and collective dynamics of fluids, plasmas and beams are presented. In particular, the pictures given by the Madelung fluid and the Moyal-Ville-Wigner phase-space quasidistribution, including the related quantum tools such as marginal distributions for the tomographic representations, are described. Some relevant applications to soliton and modulational instability theory are presented.
Although the quantum theory of the optical response of individual atoms to coherent light with frequencies close to electronic transitions and the fluid equations for a gas are well known and understood from first principles, they are developed independently of each other and therefore cannot be applied directly to describe many of the quantum collective and transport phenomena that occur in cold atomic gases, especially in what regards their interaction with optical pulses and beams. Few attempts have been made to derive a consistent formalism and theory that are capable to model this type of systems, and those which exist rely on the adaptation of several ad-hoc hypothesis and simplifications, such as space and time dependent density operators. In this paper we provide the theoretical foundations and establish a formalism capable of paving the way for the development of new simulation tools and to explore new problems in nonlinear optics out of equilibrium.
In recent decades, substantial theoretical and experimental progress was achieved in understanding the quantum nature of physical phenomena that serves as the foundation of present and future quantum technologies. Quantum correlations like the entanglement of the states of composite systems, the phenomenon of quantum discord, which captures other aspects of quantum correlations, quantum contextuality and, connected with these phenomena, uncertainty relations for conjugate variables and entropies, like Shannon and Rényi entropies, and the inequalities for spin states, like Bell inequalities, reflect the recently understood quantum properties of micro and macro systems. The mathematical methods needed to describe all quantum phenomena mentioned above were also the subject of intense studies in the end of the last, and beginning of the new, century. 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Review of different representations of signals including the phase-space representations and tomographic representations is presented. The signals under consideration are either linear or nonlinear ones. The linear signals satisfy linear quantumlike Schrödinger and von Neumann equations. Nonlinear signals satisfy nonlinear Schrödinger equations as well as Gross-Pitaevskii equation describing solitons in Bose-Einstein condensate. The Ville-Wigner distributions for solitons are considered in comparison with tomographic-probability densities describing solitons completely. different kinds of tomographies — symplectic tomography, optical tomography and Fresnel tomography are reviewed. New kind of map of the signals onto probability distributions of discrete photon number-like variable is discussed. Mutual relations between different transformations of signal functions are established in explicit form. Such characteristics of the signal-probability distribution as entropy is discussed.
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