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Zhang Ke, Fan Cheng-Yu, Fan Hong-Yi. Optical complex integration-transform for deriving complex fractional squeezing operator. Chinese Physics B, 2020, 29(3): 030306  
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Optical complex integration-transform for deriving complex fractional squeezing operator
Zhang Ke1, 2, 3, Fan Cheng-Yu1, †, Fan Hong-Yi2
Key Laboratory of Atmospheric Optics, Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Hefei 230031, China
University of Science and Technology of China, Hefei 230031, China
Huainan Normal University, Huainan 232038, China

 

† Corresponding author. E-mail: cyfan@aiofm.ac.cn

Project supported by the National Natural Science Foundation of China (Grant No. 11775208) and Key Projects of Huainan Normal University, China (Grant No. 2019XJZD04).

Abstract

We find a new complex integration-transform which can establish a new relationship between a two-mode operator’s matrix element in the entangled state representation and its Wigner function. This integration keeps modulus invariant and therefore invertible. Based on this and the Weyl–Wigner correspondence theory, we find a two-mode operator which is responsible for complex fractional squeezing transformation. The entangled state representation and the Weyl ordering form of the two-mode Wigner operator are fully used in our derivation which brings convenience.

PACS: ;03.65.-w;;42.50.-p;;63.20.-e;
Keyword:;integration-transform;;two-mode;;entangled state;;Weyl-Wigner correspondence theory;
1. Introduction

Optical transforms are very useful in optical signal analysis and optical communication. For instance, fractional Fourier transform[1] and wavelet transform[2] are widely used in optical fiber communication. These transforms can be developed in the context of quantum optics. In this paper, we shall propose a new complex integration-transform which can establish a new relationship between a two-mode operator’s matrix element in the entangled state representation (EGR) and its Wigner function. Based on this, we find an optical complex fractional squeezing transformation by virtue of the Weyl–Wigner correspondence[3–5] in the entangled state representation.[6,7] The work is arranged as follows. In Section 2, we briefly review the EGR, based on which in Section 3 we derive the two-mode Wigner operator in EGR and its Weyl ordering. Then in Section 4 we find a kind of new complex integration transformation which relates a two-mode operator’s matrix element in EGR and its Wigner function. As its application, in Section 5 we derive the two-mode fractional squeezing operator which can engender the complex fractional squeezing transformation, and in Section 6 we shall derive the complex fractional squeezing transformation. Throughout the whole paper the technique of integration within ordered product (IWOP) of operators[8,9] is fully employed.

2. The entangled state representation

In Ref. [5], the entangled state representation is proposed

which obeys the common eigenstate equations

This representation is orthogonal

and complete (using and the IWOP technique)

Notice

Equation (2) becomes

The conjugate state of |η〉 is

where |ξ〉 obeys the eigenvector equations

Since [X1 + X2, P1 − P2] = 0, |ξ〉 is the common eigenstate of bipartite’s center of mass and the relative momentum,

The |ξ〉 is complete too, i.e.,

3. The two-mode Wigner operator in entangled representation

Combining the completeness of bipartite entangled states |η〉 and |ξ〉, we can make up a new completeness relation

We have the reason to introduce the normally ordered form

and it turns out that Δ (σ,γ) is just the two-mode Wigner operator. Because when letting γ = α + β*, σ = α − β*, we can see

which is just the direct product of two single-mode Wigner operators.

The marginal distribution of the two-mode Wigner operator constructed in this way is

We can further show that the form of Δ (σ, γ) in |ξ〉 representation is

while in 〈η| representation is

Using the integration method within normally ordered product we perform the integration in Eqs. (17) and (18), which leads to Eq. (14).

4. New complex integral transformation connecting |η〉 〈η |ξ〉 〈ξ| and the two-mode Wigner operator

Recall that the single-mode Winger operator’s Weyl ordering form is

where the symbol denotes Weyl ordering, which we firstly introduced in Ref. [10].

Equation (19) is the combination of and . As its generalization, from the eigenvector equations (2) and (9) we know

so the Weyl ordering form of the two-mode Wigner operator is

The original meaning of Weyl ordering can be traced back to the operator identity , which is different from the , ordering, , where . Thus, using the Baker–Hausdorff operator formula we can convert the coordinate–momentum projecting operator into its Weyl ordering form,

Likewise, from Eq. (19) and noting , we may convert |η〉 〈η|ξ〉 〈ξ| into its Weyl ordering form, i.e.,

Using Eq. (22), we can put

Its inverse transformation is

Thus we find a new kind of complex integral transformation whose integral kernel is e−(ξ−μ)(η*−ν*) + (η−ν)(ξ*−μ*).

5. New relationship between a two-mode operator’s matrix element in the entangled state representation and its Wigner function

In this section we shall show that the complex integration-transform can establish a new relationship between a two-mode operator’s matrix element in the entangled state representation and its Wigner function.

Assuming that an operator ’s classical Weyl correspondence function is F(η,ξ), then using the Wigner operator’s entangled state representation (7), we obtain

Then we make up the above mentioned integration transform for F(η,ξ)

where |μ〉 belongs to the |ξ〉 representation, in the last step we have performed the integration over d2σ, which converts |ν−2σ〉 to its conjugate |μ〉

The inverse of Eq. (27) is

This formula shows the relationship between ’s Wigner function F(η,ξ) and its matrix element in the entangled state representation.

6. Derivation of the complex fractional squeezing transformation

When the complex classical function is

here α is an angle parameter, substituting Eq. (30) into Eq. (27) and performing this integration, we obtain

with

This is the generalization of the fractional Fourier integral kernel,[1] and is named the fractional squeezing integration transform kernel. For obtaining the fractional squeezing operator, we construct the integration

where |μ〉 and 〈μ| are mutual conjugate entangled states,

Using Eq. (32) and IWOP, we perform the integration in Eq. (33) and obtain

which is the complex fractional squeezing operator.

7. Conclusion

By using the two-mode Wigner operator in EGR and its Weyl ordering form we have found a kind of new integration transformation which relates a two-mode operator’s matrix element in EGR and its Wigner function. In this way the complex fractional squeezing operator is derived and the phase space quantum mechanics[11,12] is developed.

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