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Functional Forms for the Squeeze and the Time-Displacement Operators

Michael Martin Nieto, Universität Ulm

Published Jan 1, 1996• 14 citations
OpenAlex

Abstract

Using Baker-Campbell-Hausdorff relations, the squeeze and harmonic-oscillator time-displacement operators are given in the form \exp[\delta I] \exp[\alpha (x^2)]\exp[\beta(x\partial)] \exp[\gamma (\partial)^2], where \alpha, \beta, \gamma, and \delta are explicitly determined. Applications are discussed.

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Physical Review A•Jun 1, 1988•13 citations
In a recent paper (Nieto M M 1996 Quantum Semiclass. Opt. 8 1061, quant-ph/9605032), the one-dimensional squeezed and harmonic oscillator time-displacement operators were reordered in coordinate-momentum space. In this paper, we give a general approach for reordering the multidimensional exponential quadratic operator (EQO) in coordinate-momentum space. An explicit computational formula is provided and applied to the single-mode and double-mode EQO through the squeezed operator and the time-displacement operator of the harmonic oscillator.
We obtain the wave function in the coordinate representation of the one-mode squeezed states \ensuremath{\Vert}\ensuremath{\zeta}, \ensuremath{\alpha}〉=exp(\ensuremath{\zeta}${\mathit{a}}^{\mathrm{^}2}$-${\mathrm{\ensuremath{\zeta}}}^{\mathrm{*}}$a2{\mathit{a}}^{2}a2)exp(\ensuremath{\alpha}${\mathit{a}}^{\mathrm{^}}$-${\mathrm{\ensuremath{\alpha}}}^{\mathrm{*}}$a)\ensuremath{\Vert}0〉. The wave function is a displaced Gaussian with center and width depending upon the parameters \ensuremath{\alpha} and \ensuremath{\zeta}.