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Frontiers of Physics

On the entangled fractional squeezing transformation

Hong-Yi Fan, Junhua Chen, Pengfei Zhang

Published Jan 5, 2015• 7 citations
Publisher

Abstract

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.

DOI
10.1007/s11467-014-0457-6

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We propose an entangled fractional squeezing transformation (EFrST) generated by using two mutually conjugate entangled state representations with the following operator: e−iα(a1†a2†+a1a2)eiπa2†a2e^{ - i\alpha \left( {a_1^\dag a_2^\dag + a_1 a_2 } \right)} e^{i\pi a_2^\dag a_2 }e−iα(a1†​a2†​+a1​a2​)eiπa2†​a2​ ; this transformation sharply contrasts the complex fractional Fourier transformation produced by using e−iα(a1†a1+a2†a2)eiπa2†a2e^{ - i\alpha \left( {a_1^\dag a_1 + a_2^\dag a_2 } \right)} e^{i\pi a_2^\dag a_2 }e−iα(a1†​a1​+a2†​a2​)eiπa2†​a2​ (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.
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.
Based on the fact that the quantum mechanical version of Hankel transform kernel (the Bessel function) is just the transform between | q , r 〉 and ( s , r ′|, two induced entangled state representations are given, and working with them we derive fractional squeezing-Hankel transform (FrSHT) caused by the operator e − i α ( a 1 † a 2 † + a 1 a 2 ) e − i π a 2 † a 2 , which is an entangled fractional squeezing transform operator. The additive rule of the FrSHT can be explicitly proved.
By converting the triangular functions in the integration kernel of the fractional Fourier transformation to the hyperbolic function, i.e., tan α → tanh α , sin α → sinh α , we find the quantum mechanical fractional squeezing transformation (FrST) which satisfies additivity. By virtue of the integration technique within the ordered product of operators (IWOP) we derive the unitary operator responsible for the FrST, which is composite and is made of e i πa † a /2 and exp . The FrST may be implemented in combinations of quadratic nonlinear crystals with different phase mismatches.