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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

The time development operators for Wigner functions of harmonic oscillators with quadratic Hamiltonians

Y. Ben-Aryeh, Hashem Zoubi

Published Dec 1, 1996• 8 citations
Publisher

Abstract

Analytical expressions for the time development operators for Wigner functions of harmonic oscillators with quadratic Hamiltonians are developed by the use of group theoretical methods.
DOI
10.1088/1355-5111/8/6/001

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Grossmann and Royer have recently shown that the Wigner functions are closely related to the set of all translations and inversions of phase space. This allows the phase space representation of quantum mechanics to be constructed directly on the group of phase space translations and inversions. Starting from this observation, we have derived analytical expressions for the matrix elements of the translation and inversion operators, in the harmonic oscillator representation, without introducing coordinate or momentum wavefunctions.
The Wigner distribution function (WDF) for the time-dependent quadratic Hamiltonian system is investigated in the squeezed Schrödinger cat states with the use of Lewis–Riesenfeld theory of invariants. The nonclassical aspects of the system produced by superposition of two distinct squeezed states are analyzed with emphasis on their application into special systems beyond simple harmonic oscillator. An application of our development to the measurement of quantum state by reconstructing the WDF via Autler–Townes spectroscopy is addressed. In addition, we considered particular models such as Cadirola–Kanai oscillator, frequency stable damped harmonic oscillator, and harmonic oscillator with time-variable frequency as practical applications with the object of promoting the understanding of nonclassical effects associated with the WDF.