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Quantum Optics Journal of the European Optical Society Part B

The higher-order squeezing for one- and two-mode combination squeezed states

Zhongxi Zhang, Fan Hong‐Yi

Published Jun 1, 1993• 1 citation
Publisher

Abstract

The one- and two-mode combination squeezed states are constructed by applying the operator U identical to exp(-i lambda (erQ1P2+e-rQ2P1)) or W identical to exp(-i lambda (erQ1Q2-e-rP1P2)) (where lambda , r are real; Qi, Pi, i=1, 2 are coordinate and momentum operators, respectively) to the vacuum state. When the condition tanh lambda <1/(1+cosh r) is satisfied, such states can have less fluctuations in one quadrature phase than the usual two-mode squeezed state. Here, the authors examine the higher-order squeezing properties for such states. The result shows that under the same conditions, these states can exhibit a stronger form of higher-order squeezing than the usual two-mode squeezed state.

DOI
10.1088/0954-8998/5/3/003

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The one- and two-mode combination squeezed states are constructed by applying the operator U identical to exp(-i lambda (erQ1P2+e-rQ2P1)) or W identical to exp(-i lambda (erQ1Q2-e-rP1P2)) (where lambda , r are real; Qi, Pi, i=1, 2 are coordinate and momentum operators, respectively) to the vacuum state. When the condition tanh lambda <1/(1+cosh r) is satisfied, such states can have less fluctuations in one quadrature phase than the usual two-mode squeezed state. Here, the authors examine the higher-order squeezing properties for such states. The result shows that under the same conditions, these states can exhibit a stronger form of higher-order squeezing than the usual two-mode squeezed state.
We find that the exponential operators U=exp[-i(${\ensuremath{\lambda}}_{1}$Q1{\mathit{Q}}_{1}Q1​P2{\mathit{P}}_{2}P2​+${\ensuremath{\lambda}}^{2}$Q2{\mathit{Q}}_{2}Q2​P1{\mathit{P}}_{1}P1​] and W=exp[-i(${\ensuremath{\lambda}}_{1}$Q1{\mathit{Q}}_{1}Q1​Q2{\mathit{Q}}_{2}Q2​-${\ensuremath{\lambda}}_{2}$pP1{\mathrm{P}}_{1}P1​P2{\mathit{P}}_{2}P2​)] (where ${\ensuremath{\lambda}}_{1}$ and ${\ensuremath{\lambda}}_{2}$ are real; Pi{P}_{i}Pi​ and Qi{Q}_{i}Qi​ 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.
By extending the usual two-mode squeezing operator S2 = exp [iλ (Q1P2 +Q2P1)] to the three-mode squeezing operator S3 = exp {iλ [Q1 (P2 + P3) +Q2 (P1 + P3) +Q3 (P1 + P2)]}, we obtain the corresponding three-mode squeezed coherent state. The state’s higher-order properties, such as higher-order squeezing and higher-order sub-Possonian photon statistics, are investigated. It is found that the new squeezed state not only can be squeezed to all even orders but also exhibits squeezing enhancement comparing with the usual cases. In addition, we examine the violation of Bell-inequality for the three-mode squeezed states by using the formalism of Wigner representation. PACS: 03.65.-W; 03.65.Ud
A new generalized two-mode squeezed state which has less fluctuation in one quadrature phase then the usual two-mode squeezed state at the expense of increased fluctuation in the other quadrature is proposed. The coordinate representations and normally ordered expansions of the generalized two-mode squeeze operators and are given. The wavefunction of the generalized two-mode squeezed coherent state is obtained. The non-classical properties of the state such as squeezing and sub-Poissionian statistics are discussed.
We find that the exponential operator V ≡exp[ i λ( Q 1 P 2 + Q 2 P 3 +⋅⋅⋅+ Q n −1 P n + Q n P 1 )], Q i , P i are, respectively, the coordinate and momentum operators, is an n -mode squeezing operator which engenders standard squeezing. By virtue of the technique of integration within an ordered product of operators we derive V 's normally ordered expansion and obtain the n -mode squeezed vacuum states, its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.
By extending the usual two-mode squeezing operator S2 = exp[iλ(Q1P2 + Q2P1)] to the three-mode squeezing operator S3 = exp{iλ[Q1(P2 + P3) + Q2(P1 + P3) + Q3(P1 + P2)]}, we obtain the corresponding three-mode squeezed coherent state. The higher order properties of this state, such as higher order squeezing and higher order sub-Possonian photon statistics, are investigated. It is found that the new squeezed state not only can be squeezed to all even orders but also exhibits squeezing enhancement compared with the usual cases. In addition, we examine the violation of the Bell inequality for the three-mode squeezed states by using the formalism of Wigner representation.