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Journal of the Optical Society of America A

New complex integration transformation and its compatibility with complex Weyl-Wigner transformation and entangled state representation

Hong-Yi Fan, Cui-hong Lv

Published Oct 9, 2009• 4 citations
Publisher

Abstract

Enlightened by the special transformation in our preceding paper [J. Mod. Opt. 56, 1227 (2009)], we propose a new complex integration transformation corresponding to two mutually conjugate two-mode entangled states <eta| and <xi| that is compatible with eta-xi phase space quantum mechanics and can be used to obtain the complex fractional Fourier transformation kernel from the chirplet function. This transformation obeys the Parseval theorem and is invertible.

DOI
10.1364/josaa.26.002306

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Enlightened by the special transformation in our preceding paper [J. Mod. Opt. 56, 1227 (2009)], we propose a new complex integration transformation corresponding to two mutually conjugate two-mode entangled states <eta| and <xi| that is compatible with eta-xi phase space quantum mechanics and can be used to obtain the complex fractional Fourier transformation kernel from the chirplet function. This transformation obeys the Parseval theorem and is invertible.
Based on the two mutually conjugate entangled state representations |ξ⟩ and |η⟩, we propose an integration transformation in ξ — η phase space , and its inverse transformation, which possesses some well-behaved transformation properties, such as being invertible and the Parseval theorem. This integral transformation is a convolution, where one of the factors is fixed as a special normalized exponential function. We generalize this transformation to a quantum mechanical case and apply it to studying the Weyl ordering of bipartite operators, regarding to ( Q 1 – Q 2 ) ↔ ( P 1 – P 2 ) ordered and simultaneously ( P 1 + P 2 ) ↔ ( Q 1 + Q 2 ) ordered operators.
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.
We find a new kind of integration transformation in quantum optics,which can reflect the Baker-Hausdorff formula as well as establish the transformation between phase space basis and the function space of entangled state.This transformation is invertible and satisfies Parseval-like theorem.Combining Wigner transformation and this new transformation in phase space,We can acquire a new relation between phase space function h(p,q) and corresponding Weyl-Wigner operator,which can benefit the development of quantum phase space theory.
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.