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Frontiers of Physics

3D entangled fractional squeezing transformation and its quantum mechanical correspondence

Fang Jia, Shuang Xu, Chengzhi Deng, Cunjin Liu, Liyun Hu

Published Jan 5, 2016• 5 citations

Abstract

A new type of entangled fractional squeezing transformation (EFrST) has been theoretically proposed for 2D entanglement [ Front. Phys . 10, 100302 (2015)]. In this paper, we shall extend this case to that of 3D entanglement by introducing a type of three-mode entangled state representation, which is not the product of three 1D cases. Using the technique of integration within an ordered product of operators, we derive a compact unitary operator corresponding to the 3D fractional entangling transformation, which is an entangling operator that presents a clear transformation relation. We also verified that the additivity property of the novel 3D EFrST is of a Fourier character by using its quantum mechanical description. As an application of this representation, the EFrST of the three-mode number state is calculated using the quantum description of the EFrST.

DOI: 10.1007/s11467-015-0538-1

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A new type of entangled fractional squeezing transformation (EFrST) has been theoretically proposed for 2D entanglement [ Front. Phys . 10, 100302 (2015)]. In this paper, we shall extend this case to that of 3D entanglement by introducing a type of three-mode entangled state representation, which is not the product of three 1D cases. Using the technique of integration within an ordered product of operators, we derive a compact unitary operator corresponding to the 3D fractional entangling transformation, which is an entangling operator that presents a clear transformation relation. We also verified that the additivity property of the novel 3D EFrST is of a Fourier character by using its quantum mechanical description. As an application of this representation, the EFrST of the three-mode number state is calculated using the quantum description of the EFrST.

We propose an entangled fractional squeezing transformation (EFrST) generated by using two mutually conjugate entangled state representations with the following operator:

e^{ - i\alpha \left( {a_1^\dag a_2^\dag + a_1 a_2 } \right)} e^{i\pi a_2^\dag a_2 }

; this transformation sharply contrasts the complex fractional Fourier transformation produced by using

e^{ - i\alpha \left( {a_1^\dag a_1 + a_2^\dag a_2 } \right)} e^{i\pi a_2^\dag a_2 }

(see Front. Phys. DOI: 10.1007/s11467-014-0445-x). The EFrST is obtained by converting the triangular functions in the integration kernel of the usual fractional Fourier transformation into hyperbolic functions, i.e., tanα → tanhα and sinα → sinh α. The fractional property of the EFrST can be well described by virtue of the properties of the entangled state representations.

In this Letter, a new fractional entangling transformation(Fr ET) is proposed, which is generated in the entangled state representation by a unitary operator expfiθeabt abTg where aebT is the Bosonic annihilate operator. The operator is actually an entangled one in quantum optics and differs evidently from the separable operator, expfiθeaa t bbTg, of complex fractional Fourier transformation. The additivity property is proved by employing the entangled state representation and quantum mechanical version of the Fr ET. As an application, the Fr ET of a two-mode number state is derived directly by using the quantum version of the Fr ET, which is related to Hermite polynomials.

In this Letter, a new fractional entangling transformation (FrET) is proposed, which is generated in the entangled state representation by a unitary operator expfiθðab † þ a † bÞg where aðbÞ is the Bosonic annihilate operator.The operator is actually an entangled one in quantum optics and differs evidently from the separable operator, expfiθða † a þ b † bÞg, of complex fractional Fourier transformation.The additivity property is proved by employing the entangled state representation and quantum mechanical version of the FrET.As an application, the FrET of a two-mode number state is derived directly by using the quantum version of the FrET, which is related to Hermite polynomials.