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Wanxi Xueyuan xuebao

Dirac's Symbolic Notation of Squeezed Displacement Operator and Its Application

Song Jun

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

This paper presents Dirac's symbolic notation of a real parameter's squeezed displacement operator,which normal product has been derived by virtue of the technique of integration within an ordered product(IWOP) of operators,and two application examples are also given.

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Via the route of developing Dirac's symbolic method by virtue of the technique of integration within an ordered product (IWOP) of operators we derive various entangled state representations, squeezed states, and operator ordering formulae. We show how the entangled states can be applied to many aspects of quantum optics theory. The applications of the entangled states in the interdisciplinary study of quantum optics and Fourier optics are also introduced. All these discussions exhibit the power of Dirac's symbolic method with the aid of the IWOP technique.
We present a general unified approach for arranging quantum operators of optical fields into ordered products (normal ordering, antinormal ordering, Weyl ordering (or symmetric ordering)) by fashioning Dirac's symbolic method and representation theory. We propose the technique of integration within an ordered product (IWOP) of operators to realize our goal. The IWOP makes Dirac's representation theory and the symbolic method more transparent and consequently more easily understood. The beauty of Dirac's symbolic method is further revealed. Various applications of the IWOP technique, such as in developing the entangled state representation theory, nonlinear coherent state theory, Wigner function theory, etc, are presented.
It is known that exp[i\lamda(Q_1P_1-i/2)] is a unitary single-mode squeezing operator, where Q_1,P_1 are the coordinate and momentum operators, respectively. In this paper we employ Dirac's coordinate representation to prove that the exponential operator S_{n}=Exp[i\lamda sum_{i=1}^{n}](Q_{i}P_{i+1}+Q_{i+1}P_{i}))], (Q_{n+1}=Q_1P_{n+1}=P_1), is a n-mode squeezing operator which enhances the standard squeezing. By virtue of the technique of integration within an ordered product of operators we derive S_{n}'s normally ordered expansion and obtain new n-mode squeezed vacuum states, its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.