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Physica Scripta

New approach for obtaining the squeezing-enhanced state and its Wigner function by virtue of the Weyl–Wigner quantization scheme

Hong-Yi Fan, Hong‐Chun Yuan, Xue-xiang Xu, Nian-Quan Jiang

Published Jan 1, 2011• 8 citations
Publisher

Abstract

By virtue of the Weyl?Wigner quantization scheme, we derive the explicit form of the squeezing-enhanced state without disentangling the generalized squeezing operator where Q and P are two quadratures of the light field; this approach is convenient and concise. By virtue of the Weyl-ordering invariance under similar transformations, the Wigner function of the squeezing-enhanced state is derived and its figure is depicted, which clearly exhibits the squeezing enhancement.

DOI
10.1088/0031-8949/83/01/015403

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By virtue of the Weyl?Wigner quantization scheme, we derive the explicit form of the squeezing-enhanced state without disentangling the generalized squeezing operator where Q and P are two quadratures of the light field; this approach is convenient and concise. By virtue of the Weyl-ordering invariance under similar transformations, the Wigner function of the squeezing-enhanced state is derived and its figure is depicted, which clearly exhibits the squeezing enhancement.
Squeezed state is theoretically obtained by performing the squeezing operator on an initial state.Due to non-commutable operators each other,it is not easy to directly obtain the explicit expression of squeezed state.Here,a general approach is provided for deriving the squeezed state′s explicit form,namely,by virtue of the Weyl-Wigner correspondence scheme the explicit forms of single-mode and two-mode squeezed state is obtained,respectively.Without disentangling the squeezing operator,this adopted method is applicable to explicitly solve the general multimode case.
By virtue of the property that Weyl ordering is invariant under similar transformations we show that the Weyl ordered form of the Wigner operator, a Dirac δ-operator function, brings much convenience for deriving miscellaneous Wigner transforms. The operators which engender various transforms of the Wigner operator, can also be easily deduced by virtue of the Weyl ordering technique. The correspondence between the optical Wigner transforms and the squeezing transforms in quantum optics is investigated.
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.
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[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.