Laguerre-Gaussian Modes and the Wigner Transform

Recent developments in laser physics have called renewed attention to Laguerre-Gaussian (LG) beams of paraxial light. In this paper we consider the corresponding LG modes for the two-dimensional harmonic oscillator, which appear in the transversal plane at the laser beam's waist. We see how they arise as Wigner transforms of Hermite-Gaussian modes, and we proceed to find a closed form for their own Wigner transforms, providing an alternative to the methods of Simon and Agarwal. Our main observation is that the Wigner transform intertwines the creation and annihilation operators for the two classes of modes.Comment: 12 pages, minor corrections; submitted, Journal of Modern Optic

Manipulation of Semiclassical Photon States

Gabriel F. Calvo and Antonio Picon defined a class of operators, for use in quantum communication, that allows arbitrary manipulations of the three lowest two-dimensional Hermite-Gaussian modes {|0,0>,|1,0>,|0,1>}. Our paper continues the study of those operators, and our results fall into two categories. For one, we show that the generators of the operators have infinite deficiency indices, and we explicitly describe all self-adjoint realizations. And secondly we investigate semiclassical approximations of the propagators. The basic method is to start from a semiclassical Fourier integral operator ansatz and then construct approximate solutions of the corresponding evolution equations. In doing so, we give a complete description of the Hamilton flow, which in most cases is given by elliptic functions. We find that the semiclassical approximation behaves well when acting on sufficiently localized initial conditions, for example, finite sums of semiclassical Hermite-Gaussian modes, since near the origin the Hamilton trajectories trace out the bounded components of elliptic curves.Comment: 30 pages, 3 figures. Small corrections, mostly in Section V. To appear in the Journal of Mathematical Physic

Open science in archaeology

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