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Abstract
Two fractional Fourier transforms are used to define bi-fractional displacement operators, which interpolate between displacement operators and parity operators. They are used to define bi-fractional coherent states. They are also used to define the bi-fractional Wigner function, which is a two-parameter family of functions that interpolates between the Wigner function and the Weyl function. Links to the extended phase space formalism are also discussed.
Highlights
Wigner and Weyl functions and P and Q functions play a central role in the general area of phase space methods[1, 2]
The displacement and parity operators are related through a two-dimensional Fourier transform, which we replace with fractional Fourier transforms [4, 5, 6, 7]
This leads to a two-parameter family of operators which we call bi-fractional displacement operators, and which include as special cases the displacement and parity operators
Summary
Wigner and Weyl functions and P and Q functions play a central role in the general area of phase space methods[1, 2]. In the present paper we review briefly and extend this work, by discussing links with the extended phase space method in refs[8, 9, 10, 11, 12] The latter uses the fact that the Wigner function W (x, p) is related to the Weyl function W (X, P ) through a two-dimensional Fourier transform. An ‘extended Wigner function’ (and other ‘extended quantities’) that depend on x, p, X, P have been introduced in the extended phase space x − p − X − P In this language, the two variables of the bi-fractional displacement operator studied here, are in the direction defined by the angle φ1 in the x − P plane, and in the direction defined by the angle φ2 in the p − X plane (see fig).
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