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arXiv:0902.1589 · Europhysics Letters (EPL)

New n-mode squeezing operator and squeezed states with standard squeezing

Li-yun Hu, Hong-yi Fan

Published Mar 1, 2009• 14 citations
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Abstract

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.

DOI
10.1209/0295-5075/85/60001

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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.
It is known that exp [iλ ( Q 1 P 1 – i/2)] is a unitary single-mode squeezing operator, where Q 1 , P 1 are the coordinate and momentum operators, respectively. In this paper we employ Dirac's coordinate representation to prove that the exponential operator , is an n -mode squeezing operator which enhances the standard squeezing. By virtue of the technique of integration within an ordered product of operators we derive S n 's normally ordered expansion and obtain new n -mode squeezed vacuum states, its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.
It is known that exp[i\lamda(Q_1P_1-i/2)] is a unitary single-mode squeezing operator, where Q_1,P_1 are the coordinate and momentum operators, respectively. In this paper we employ Dirac's coordinate representation to prove that the exponential operator S_{n}=Exp[i\lamda sum_{i=1}^{n}](Q_{i}P_{i+1}+Q_{i+1}P_{i}))], (Q_{n+1}=Q_1P_{n+1}=P_1), is a n-mode squeezing operator which enhances the standard squeezing. By virtue of the technique of integration within an ordered product of operators we derive S_{n}'s normally ordered expansion and obtain new n-mode squeezed vacuum states, its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.
By virtue of the technique of integration within an ordered product of operators a new four-mode squeezing operator that squeezes the four-mode quadrature operators of light field in the standard way is found. This operator differs from the direct product of two two-mode squeezing operators. It is the exponential operator V ≡exp [ i r ( Q 1 P 2 + Q 2 P 3 + Q 3 P 4 + Q 4 P 1 )]. The Wigner function of the new four-mode squeezed state is calculated, which quite differs from that of the direct-product state of two usual two-mode squeezed states.
In a 3-wave mixing process occurring in some nonlinear optical medium when }a1†a_{1}^{\dagger}a1†​{\small mode interacts with both }$a_{2}^{\dagger}% modeand{\small mode and }modeanda_{3}^{\dagger}mode,wetheoreticallystudythesqueezingeffectgeneratedbytheoperator{\small mode, we theoretically study the squeezing effect generated by the operator }mode,wetheoreticallystudythesqueezingeffectgeneratedbytheoperatorS_{3}\equiv \exp[μ(a_{1}% a_{2}-a_{1}^{\dagger}a_{2}^{\dagger})+ν(a_{1}a_{3}-a_{1}^{\dagger}% a_{3}^{\dagger})].Thenew3−modesqueezedvacuumstateinFockspaceisderived,andtheuncertaintyrelationforitisdemonstrated,Itturnsoutthat{\small . The new 3-mode squeezed vacuum state in Fock space is derived, and the uncertainty relation for it is demonstrated, It turns out that }.Thenew3−modesqueezedvacuumstateinFockspaceisderived,andtheuncertaintyrelationforitisdemonstrated,ItturnsoutthatS_{3}mayexhibitenhancedsqueezing.Byvirtueofthetechniqueofintegrationwithinanorderedproduct(IWOP)ofoperators,wealsoderive{\small may exhibit enhanced squeezing. By virtue of the technique of integration within an ordered product (IWOP) of operators, we also derive }mayexhibitenhancedsqueezing.Byvirtueofthetechniqueofintegrationwithinanorderedproduct(IWOP)ofoperators,wealsoderiveS_{3}${\small 's normally ordered expansion. The Wigner function of new 3-mode squeezed vacuum state is calculated by using the Weyl ordering invariance under similar transformations.
One- and two-mode successively squeezed state, obtained through re-squeezing two single mode squeezed states by the two-mode squeezing operator, is studied in terms of the technique of integration within an ordered product (IWOP) of operators. We first derive the normally ordered form of this one- and two-mode successively squeezing operator, and then investigate the quantum statistical properties of the corresponding squeezed state. Particularly, we use the Weyl ordering invariance under a similar transformation to derive the analytical expression of its Wigner function, which seems very easy and concise. Finally, the experimental generation of one- and two-mode successively squeezed state is also proposed simply.