Interpolation between phase space quantities with bifractional displacement operators

TL;DR

This paper introduces bifractional displacement operators and associated phase space functions, providing a new framework for interpolating between different phase space representations using coherent states and generalized Wigner, Q, and P functions.

Contribution

It presents the novel concept of bifractional displacement operators and phase space functions, expanding the toolkit for quantum state analysis and interpolation between phase space quantities.

Findings

Defined bifractional displacement operators as special cases of group G.

Constructed bifractional coherent states from vacuum using these operators.

Introduced bifractional Wigner, Q, and P functions that interpolate between known phase space functions.

Abstract

Bifractional displacement operators, are introduced by performing two fractional Fourier transforms on displacement operators. They are shown to be special cases of elements of the group G, that contains both displacements and squeezing transformations. Acting with them on the vacuum we get various classes of coherent states, which we call bifractional coherent states. They are special classes of squeezed states which can be used for interpolation between various quantities in phase space methods. Using them we introduce bifractional Wigner functions A(?, ?; ??, ??), which are a two-dimensional continuum of functions, and reduce to Wigner and Weyl functions in special cases. We also introduce bifractional Q-functions, and bifractional P-functions. The physical meaning of these quantities is discussed.