Physical Review A

Coordinate representation of squeezed states

Jagdish Rai, C. L. Mehta

Published Jun 1, 1988• 13 citations

Abstract

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{*}}$${\mathit{a}}^{2}$)exp(\ensuremath{\alpha}${\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}.

DOI: 10.1103/physreva.37.4497

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Physical Review A

Coordinate representation of squeezed states

Jagdish Rai, C. L. Mehta

Published Jun 1, 1988• 13 citations

Publisher

Abstract

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{*}}$${\mathit{a}}^{2}

)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}.

DOI: 10.1103/physreva.37.4497

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Coordinate wave functions for the one-mode squeezed states produced by the quantum analog of the general linear transformation in phase space are calculated. The probability density [\ensuremath{\Vert}\ensuremath{\psi}(q)${\ensuremath{\Vert}}^{2}$] for these states is Gaussian with center predicted by the classical transformation. The quantum image (which includes the traditional two-mode squeeze operator) of a three-parameter symplectic map in two-mode phase space equally generates squeezed states having Gaussian \ensuremath{\Vert}\ensuremath{\psi}( ${q}_{1}$ , ${q}_{1}$ )${\ensuremath{\Vert}}^{2}$. The center of the two-mode Gaussian is again predicted by the classical mapping.Coordinate wave functions for the one-mode squeezed states produced by the quantum analog of the general linear transformation in phase space are calculated. The probability density [\ensuremath{\Vert}\ensuremath{\psi}(q)${\ensuremath{\Vert}}^{2}$] for these states is Gaussian with center predicted by the classical transformation. The quantum image (which includes the traditional two-mode squeeze operator) of a three-parameter symplectic map in two-mode phase space equally generates squeezed states having Gaussian \ensuremath{\Vert}\ensuremath{\psi}(${q}_{1}$,${q}_{1}$)${\ensuremath{\Vert}}^{2}$. The center of the two-mode Gaussian is again predicted by the classical mapping.

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arXiv:0903.0997Mar 5, 20090 citations

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{*}}$${\mathit{a}}^{2}

)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}.

{q}_{1}

{q}_{1}

)${\ensuremath{\Vert}}^{2}$. The center of the two-mode Gaussian is again predicted by the classical mapping.

We find that the exponential operators U=exp[-i(${\ensuremath{\lambda}}_{1}$

{\mathit{Q}}_{1}

{\mathit{P}}_{2}

+${\ensuremath{\lambda}}^{2}$

{\mathit{Q}}_{2}

{\mathit{P}}_{1}

] and W=exp[-i(${\ensuremath{\lambda}}_{1}$

{\mathit{Q}}_{1}

{\mathit{Q}}_{2}

-${\ensuremath{\lambda}}_{2}$p

{\mathrm{P}}_{1}

{\mathit{P}}_{2}

)] (where ${\ensuremath{\lambda}}_{1}$ and ${\ensuremath{\lambda}}_{2}$ are real;

{P}_{i}

and

{Q}_{i}

are coordinate and momentum operators, respectively; and i=1,2) are two types of generalized squeezing operators responsible for generating two types of one- and two-mode combination squeezed states, respectively. The coordinate and/or momentum representations of U and W are presented, and their normal product forms are first derived in terms of the newly developed technique of integration within an ordered product, which provides us with a direct approach to constructing these new squeezed states. The fluctuations in quadrature phases for these states are analyzed.