Physical Review A

Representations of two-mode squeezing transformations

Fan Hong-Yi, Fan Yue

Published Jul 1, 1996• 169 citations

Abstract

We derive natural representations for two-mode squeezing operators. By natural we mean that the representations are composed by the eigenvectors of either one of the two two-mode quadrature operators for squeezing. The technique of integration within an ordered product of operators provides us with a convenient method of derivation. \textcopyright{} 1996 The American Physical Society.

DOI: 10.1103/physreva.54.958

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Physical Review A

Representations of two-mode squeezing transformations

Fan Hong-Yi, Fan Yue

Published Jul 1, 1996• 169 citations

Publisher

Abstract

DOI: 10.1103/physreva.54.958

Related papers

Squeezed states: Operators for two types of one- and two-mode squeezing transformations

Fan Hong-Yi

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.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.

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We show that the best way to study the relationship between quantum entanglement and quantum squeezing for the multimode case is through constructing the new n -pair entangled state representation . The explicit form of the 2 n -mode squeezing operator which squeezes the n -pair entangled state can be directly derived via the transition from to where M is an n × n complex matrix, and the technique of integration within an ordered product of operators. We also analyze the squeezing properties of the 2 n -mode squeezed state for M being Hermitian, and obtain the variances of the 2 n -mode quadrature operators in the 2 n -mode squeezed vacuum state with a concise result; the condition for reaching the minimum of the uncertainty relationship is also investigated.We show that the best way to study the relationship between quantum entanglement and quantum squeezing for the multimode case is through constructing the new n -pair entangled state representation . The explicit form of the 2 n -mode squeezing operator which squeezes the n -pair entangled state can be directly derived via the transition from to where M is an n × n complex matrix, and the technique of integration within an ordered product of operators. We also analyze the squeezing properties of the 2 n -mode squeezed state for M being Hermitian, and obtain the variances of the 2 n -mode quadrature operators in the 2 n -mode squeezed vacuum state with a concise result; the condition for reaching the minimum of the uncertainty relationship is also investigated.

Journal of Physics A Mathematical and TheoreticalMay 26, 20103 citations

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.