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arXiv:nlin/0207057

Tomography of Solitons

Sergio M. De Nicola, Renato Fedele, Margarita A. Man’ko, Vladimir I. Man’ko

Published Jul 30, 2002• 0 citations
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Abstract

We develope the tomographic representation ofwavefunctions which are solutions of the generalizednonlinear Schrodinger equation (NLSE) and show itsconnection with the Weyl--Wigner map.In particular, this theory is applied tothe envelope solitons, where tomograms for envelopebright solitons of a wide family of modified NLSE arepresented and numerically evalueted.

DOI
10.48550/arxiv.nlin/0207057

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We develope the tomographic representation ofwavefunctions which are solutions of the generalizednonlinear Schrodinger equation (NLSE) and show itsconnection with the Weyl--Wigner map.In particular, this theory is applied tothe envelope solitons, where tomograms for envelopebright solitons of a wide family of modified NLSE arepresented and numerically evalueted.
We develop the tomographic representation of wavefunctions which are solutions of the generalized nonlinear Schrödinger equation (NLSE) and show its connection with the Weyl–Wigner map. The generalized NLSE is presented in the form of a nonlinear Fokker–Planck-type equation for the standard probability distribution function (tomogram). In particular, this theory is applied to solitons, where tomograms for envelope bright solitons of a family of modified NLSEs are presented and numerically evaluated. Examples of symplectic tomography and Fresnel tomography of linear and nonlinear signals are discussed.
Some non-linear generalizations of classical Radon tomography were recently introduced by M. Asorey et al [Phys. Rev. A 77, 042115 (2008), where the straight lines of the standard Radon map are replaced by quadratic curves (ellipses, hyperbolas, circles) or quadratic surfaces (ellipsoids, hyperboloids, spheres). We consider here the quantum version of this novel non-linear approach and obtain, by systematic use of the Weyl map, a tomographic encoding approach to quantum states. Non-linear quantum tomograms admit a simple formulation within the framework of the star-product quantization scheme and the reconstruction formulae of the density operators are explicitly given in a closed form, with an explicit construction of quantizers and dequantizers. The role of symmetry groups behind the generalized tomographic maps is analyzed in some detail. We also introduce new generalizations of the standard singular dequantizers of the symplectic tomographic schemes, where the Dirac delta-distributions of operator-valued arguments are replaced by smooth window functions, giving rise to the new concept of "thick" quantum tomography. Applications for quantum state measurements of photons and matter waves are discussed.
The Wigner transformation is used to define the quasidistribution(Wigner function) associated with the wave function of the cylindrical nonlinear Schrodinger equation (CNLSE) in a way similar to that of the standard nonlinear Schr¨odinger equation (NLSE). The phase-space equation, governing the evolution of such quasidistribution, is a sort of nonlinear von Neumann equation (NLvNE), called here the ‘cylindrical nonlinear von Neumann equation’ (CNLvNE). Furthermore, the phase-space transformations, connecting the Wigner function and the NLvNE with the ‘cylindrical Wigner function’ and the CNLvNE, are found by extending the configuration space transformations that connect the NLSE and the CNLSE. Some examples of phase-space soliton solutions are given analytically andevaluated numerically.
The Wigner function, a kind of wavelet transform is studied for one- and two-soliton solutions to the nonlinear Schrödinger equation. The exact expression of the Wigner transform for the one-soliton solution is obtained. The features of soliton-soliton interaction of different two-soliton states, the states of coherence analogous to that of two-localized wave packets in quantum mechanics, are displayed. Suitability of this transform for analyzing “ soliton ” in open systems is also discussed.