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Chinese Physics B

From fractional Fourier transformation to quantum mechanical fractional squeezing transformation

Cui-hong Lv, Hong-Yi Fan, Dongwei Li

Published Feb 1, 2015• 11 citations
Publisher

Abstract

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.

DOI
10.1088/1674-1056/24/2/020301

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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.
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.
Fourier and fractional-Fourier transformations are widely used in theoretical physics. In this paper we make quantum perspectives and generalization for the fractional Fourier transformation (FrFT). By virtue of quantum mechanical representation transformation and the method of integration within normal ordered product (IWOP) of operators, we find the key point for composing FrFT, and reveal the structure of FrFT. Following this procedure, a full family of generalized fractional transformations are discovered with the usual FrFT as one special case. The eigen-functions of arbitrary GFrT are derived explicitly.
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.
Starting from the optical fractional Fourier transform (FFT) and using the technique of integration within an ordered product of operators we establish a formalism of FFT for quantum mechanical wave functions. In doing so, the essence of FFT can be seen more clearly, and the FFT of some wave functions can be derived more directly and concisely. We also point out that different FFT integral kernels correspond to different quantum mechanical representations. They are generalized FFT. The relationship between the FFT and the rotated Wigner operator is studied by virtue of the Weyl ordered form of the Wigner operator.
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.
This paper presents a generalized circuit framework for constructing Shih-type fractionalizations of unitary operators of dyadic order, i.e., operators UUU satisfying U2n=IU^{2^n}=IU2n=I. Building upon the architecture of the quantum fractional Fourier transform (QFrFT), we show that fractionalization can be implemented coherently as a weighted superposition of integer powers, ∑kck(α)Uk\sum_k c_k(α)U^k∑k​ck​(α)Uk, where the coefficients are generated through an ancilla-domain quantum Fourier transform and a diagonal phase modulation. Under the assumption that controlled implementations of the required powers of UUU are available, the resulting circuit yields a parameterized family of operators that interpolates the integer powers of UUU and satisfies the additive property of fractional transforms. As concrete applications, we derive explicit quantum circuit realizations of the quantum fractional Hartley transform (QFrHT) and of the fractional cosine-transform families associated with Types~I and~IV. These constructions demonstrate the versatility of the proposed dyadic-order fractionalization framework for structured operators arising in quantum signal processing.