Using supermodels in quantum optics
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Starting from supersymmetric quantum mechanics and related supermodels within Schr¨odinger theory, we review the meaning of self-similar superpotentials which exhibit the spectrum of a geometric series. We construct special types of discretizations of the Schr¨odinger equation on time scales with particular symmetries. This discretization leads to the same type of point spectrum for the referred Schr¨odinger difference operator than in the self-similar superpotential case, hence exploiting an isospectrality situation. A discussion is opened on the question of how the considered energy sequence and its generalizations serve the understanding of coherent states in quantum optics. Copyright © 2006 N. Garbers and A. Ruffing. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Items like “coherent states” or “squeezed states” can nowadays be found in many recent articles on quantum optics. The fact that the Nobel Prize in Physics 2005 has been awarded to pioneers on this area, like R. Glauber, gives insight how active this area is. The kind of physical states behind coherent states or squeezed states are the so-called nonclassical states. They are minimal uncertainty states. These properties are essential for an efficient signal transmission in the quantum world. The theory of coherent states in physics has been developed all over the last decades, among others by Glauber, Klauder, and Sudarshan. Coherent states play a major role in laser physics. The mathematical modeling in laser physics allows three different approaches to coherent states: first by the method of translation operators, second by the method of ladder operators, and third by the method of minimal uncertainty. Nonclassical states like squeezed laser fields are very important for applications: the experimental methods when dealing with squeezed laser fields include for instance the so-called self-homodyne tomography. The mathematical modeling in selfhomodyne tomography allows a tomographical reconstruction of the Wigner function, belonging to a set of probability densities of fluctuations in different field amplitudes. Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 72768, Pages 1–14 DOI 10.1155/ADE/2006/72768
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Using supermodels in quantum optics
Squeezed laser states are—like coherent states—states of minimal uncertainty. From the viewpoint of statistics, semiconductor lasers have a super-Poisson distribution up to the pumping level, that is, a distribution which goes beyond the non-normalized Poisson distribution. But also the so-called sub-Poisson distribution has a particular meaning in multiboson systems: the definition of coherent states is directly related to solutions of Stieltjes moment problems. In [4], Penson and Solomon could show that qdiscretizations of orthogonality measures, solving the moment problems, allow to investigate mul
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