We find a new integration transformation which can convert a chirplet function to a fractional Fourier transformation kernel; this new transformation is invertible and obeys Parseval theorem. Under this transformation a new relationship between a phase space function and its Weyl–Wigner quantum correspondence operator is revealed.
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 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.
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.
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.
Fourier transforms, integer and fractional, are ubiquitous mathematical tools in basic and applied science. Certainly, since the ordinary Fourier transform is merely a particular case of a continuous set of fractional Fourier domains, every property and application of the ordinary Fourier transform becomes a special case of the fractional Fourier transform. Despite the great practical importance of the discrete Fourier transform, implementation of fractional orders of the corresponding discrete operation has been elusive. Here we report classical and quantum optical realizations of the discrete fractional Fourier transform. In the context of classical optics, we implement discrete fractional Fourier transforms of exemplary wave functions and experimentally demonstrate the shift theorem. Moreover, we apply this approach in the quantum realm to Fourier transform separable and path-entangled biphoton wave functions. The proposed approach is versatile and could find applications in various fields where Fourier transforms are essential tools.